Elastoplasticity Theory

Lecture Notes in Applied and Computational Mechanics Volume 42 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr...

Author: Koichi Hashiguchi

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Lecture Notes in Applied and Computational Mechanics Volume 42 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes of this series found on our homepage: springer.com Vol. 42: Hashiguchi K. Elastoplasticity Theory 432 p. 2009 [978-3-642-00272-4]

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Elastoplasticity Theory Koichi Hashiguchi

123

Koichi Hashiguchi 3-10-10-201, Ohtemon Fukuoka, 810-0074 Japan E-mail: [email protected]

ISBN: 978-3-642-00272-4

e-ISBN: 978-3-642-00273-1

DOI 10.1007/978-3-642-00273-1 Lecture Notes in Applied and Computational Mechanics

ISSN 1613-7736 e-ISSN 1860-0816

Library of Congress Control Number: 2009920691 © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 9876543210 springer.com

Preface Contents

Recent advancements in the performance of industrial products and structures are quite intense. Consequently, mechanical design of high accuracy is necessary to enhance their mechanical performance, strength and durability. The basis for their mechanical design can be provided through elastoplastic deformation analyses. For that reason, industrial engineers in the fields of mechanical, civil, architectural, aerospace engineering, etc. must learn pertinent knowledge relevant to elastoplasticity. Numerous books about elastoplasticity have been published since “Mathematical Theory of Plasticity”, the notable book of R. Hill (1950), was written in the middle of the last century. That and similar books mainly address conventional plasticity models on the premise that the interior of a yield surface is an elastic domain. However, conventional plasticity models are applicable to the prediction of monotonic loading behavior, but are inapplicable to prediction of deformation behavior of machinery subjected to cyclic loading and civil or architectural structures subjected to earthquakes. Elastoplasticity has developed to predict deformation behavior under cyclic loading and non-proportional loading and to describe nonlocal, finite and rate-dependent deformation behavior. The author has been lecturing on applied mechanics and has been investigating elastoplasticity for nearly a half century, during which time elastoplasticity has made great progress. Various lecture notes, research papers, review articles in English or Japanese, and books in Japanese on these subjects are piled at hand. At present, the author is continuing composition of a monograph on elastoplasticity that has been published serially in a monthly journal from June 2007, to be completed at the end of 2009, although it will unfortunately have been written entirely in Japanese. Based on those teaching and research materials, this book comprehensively addresses fundamental concepts and formulations of phenomenological elastoplasticity from the conventional to latest theories. Especially, the subloading surface model falling within the framework of the unconventional plasticity model is delineated in detail, which enables us to predict rigorously the plastic strain rate induced by the rate of stress inside the yield surface, comparing it with the other unconventional models. The viscoplastic model is also presented; it is applicable to prediction of deformation behavior in the wide range of strain rate from the quasi-static to the impact loads. Explicit constitutive equations of metals and soils are given for practical application of the theories. In addition, constitutive models of friction are described because they are indispensable for analysis of boundary

VI

Preface

value problems. Various theories proposed by the author himself are included among the contents in this book in no small number. Their detailed explanations would be possible but, on the other hand, they would easily fall into subjective explanations. For that reason, particular care was devoted to retaining objectivity in the presentation. The main purpose of this book is to expedite the application of elastoplasticity theory to analyses of engineering problems in practice. Consequently, the salient feature of this book is the exhaustive explanation of elastoplasticity, which is intended to be understood easily and clearly not only by researchers but also by beginners in the field of applied mechanics, without reading any other book. Therefore, mathematics including vector-tensor analysis and derivatives and the fundamentals in continuum mechanics are first explained to the degree necessary to understand the elastoplasticity theory described in subsequent chapters. For that reason, circumstantial explanations of physical concepts and formulations in elastoplasticity are given without a logical jump such that the derivations and transformations of all equations are described without abbreviation. Besides, a general formulation unlimited to a particular material is first addressed in detail since deformations of materials obey fundamental common characteristics, which would provide the universal knowledge for deformation of materials more than describing a formulation for a particular material. Thereafter, explicit constitutive equations of metals, soils, and friction phenomena are presented in detail, specifying material functions involved in the general formulation. Without difficulty, readers will be able to incorporate the equations included in this book into their computer programs. The author expects that a wide audience including students, engineers, and researchers of elastoplasticity will read this book and that this work will thereby contribute to the steady development of the study of elastoplasticity and applied mechanics. As a foundation, the mathematical and the physical ingredients of the continuum mechanics are treated in Chapters 1, 2, 3, and 4. Chapter 1 addresses tensor analysis. The physical quantities used in continuum mechanics are tensors; consequently, their relations are described mathematically using tensor equations. Explanations for mathematical properties and rules of tensors are presented to the extent that is sufficient to understand the subject of this book: elastoplasticity. In chapters 2–4, physical quantities for the description of deformation of solids, containing stress (rate) and strain (rate), are described with their fundamental laws. Chapter 2 is devoted to the description of motion and strain (rate) and their related quantities. Chapter 3 presents conservation laws of mass, momentum, and angular momentum, and equilibrium equations and virtual work principles derived from them. In addition, their rate forms used for constitutive equations of inelastic deformation are explained concisely. Chapter 4 specifically addresses the objectivity of constitutive equations, which is required for the description of material properties. The objectivities of various stress, strain and rotation measures are described by examining their coordinate transformation rules. Then, it is shown that the material-time derivative of state variables has no objectivity. The corotational rate tensors fulfilling the objectivity are described in detail.

Preface

VII

Chapter 5 specifically examines the description of elastic deformation. Elastic constitutive equations are classified as either hyperelasticity, Cauchy elasticity, or hypoelasticity depending on their levels of reversibility. The mathematical and physical characteristics of these equations are explained prior to the description of elastoplastic constitutive equations in the subsequent chapters. Elastoplastic constitutive equations are described comprehensively in Chapters 6–8. Chapter 6 presents the basic concepts and formulations of plastic constitutive equation, e.g. the elastic and the plastic strain rates, the consistency condition, the plastic flow rule, and the loading criterion. Then, the elastoplastic constitutive equation is formulated based on them. A description of anisotropy and the tangential inelastic strain rate are also incorporated. However, they fall within the framework of conventional plasticity on the premise that the interior of the yield surface is an elastic domain. Therefore, they are incapable of predicting a smooth transition from the elastic to plastic state and a cyclic loading behavior of real materials. In Chapter 7, the continuity and the smoothness conditions are described first. They are the fundamental requirements for the constitutive equations for irreversible deformation. Then, various unconventional elastoplastic constitutive equations are proposed, with the intention of describing the plastic strain rate induced by the rate of stress inside the yield surface. Among those equations, only the subloading surface model, assuming the subloading surface passing through the current stress point and similar to the yield surface, fulfils the requirements for elastoplastic constitutive equations: continuity and smoothness conditions. Then, the inelastic strain rate attributable to the stress rate tangential to the subloading surface is pertinently incorporated, which is indispensable for the prediction of non-proportional loading behavior observed often in plastic instability problems. In Chapter 8, the initial subloading surface model explained in Chapter 7 is shown to be incapable of describing cyclic loading behavior appropriately because it predicts only an elastic strain rate in the unloading process. Therefore, excessive strain accumulation with open hysteresis loops is predicted in the cyclic loading process. Various cyclic plasticity models have been proposed to date. Among them, only the extended subloading surface model, making a similarity center of the normal-yield and the subloading surfaces move with a plastic strain rate, can represent cyclic loading behavior appropriately, fulfilling the continuity and smoothness conditions. Notable advantages of the extended subloading surface model are presented in comparison with the other cyclic plasticity models. Chapter 9 presents a viscoplastic constitutive equation for describing ratedependent deformation induced for the stress level over the yield surface. A pertinent viscoplastic constitutive equation is described, in which the concept of the subloading surface is incorporated into the overstress model. It is applicable to the prediction of rate-dependent deformation behavior from quasi-static to impact loads. In Chapters 10 and 11, based on the elastoplastic constitutive equations described in the preceding chapters, specific constitutive equations of metals and soils are formulated. They are typical elastoplastic materials related to engineering practice. Specific yield conditions, evolution rules of hardening and softening,

VIII

Preface

anisotropy, stagnation of isotropic hardening, up/degradation of structure, etc. for these materials are incorporated in their formulations. Special issues related to elastoplastic deformation behavior are discussed in Chapters 12–14. Chapter 12 specifically examines corotational rate tensors, the necessity of which is suggested in Chapter 4. Mechanical features of corotational tensors with various spins are examined comparing their simple shear deformation characteristics. The pertinence of the plastic spin is particularly explained. Chapter 13 opens with a mechanical interpretation for the localization of deformation inducing a shear band. Then, the approaches to the prediction of shear band inception condition, inclination and thickness, e.g. eigenvalue analysis and the gradient theory are explained, and the smeared model, i.e., the shear-band embedded model for the practical finite element analysis, is described. Chapter 14 verifies first the distinguished ability of the subloading surface model for the numerical calculation in the yield state. However, it is limited to the yield state. Then, the basic equations for the return-mapping method are shown for subyield state in the subloading surface model. Chapter 15 describes the prediction of friction phenomena. All bodies except those floating in a vacuum are contacting with other bodies so that the friction phenomena occur on their surfaces. Pertinent analyses, not only of the deformation behavior of bodies but also of friction behavior on the contact surface, are necessary for the analyses of boundary-value problems. A constitutive equation of friction is formulated similarly to the elastoplastic constitutive equation. It is subsequently extended to describe the transition from a static to a kinetic friction attributable to plastic softening and the recovery of the static friction attributable to creep hardening. The author wishes to express his thanks to the colleagues at Kyushu University, who have discussed and collaborated for a long time during work undertaken until retirement: Professor M. Ueno (currently at The University of the Ryukyus), and T. Okayasu, S. Tsutsumi, and S. Ozaki (currently at Yokohama National Univ.). In addition, Professor T. Tanaka of The University of Tokyo, Professor Yatomi, C. of Kanazawa Univ., Professor F. Yoshida of Hiroshima Univ., Professor M. Kuroda of Yamagata Univ., Y. Yamakawa of Tohoku Univ., Dr. T. Ozaki of Kyushu Electric Engineering Consultants Inc., and Mr. T. Mase of Tokyo Electric Power Services Co., Ltd. are appreciated for their valuable discussions and collaborations. Furthermore, the author would like to express his sincere gratitude to Professor A. Asaoka and his colleagues at Nagoya University: Professor M. Nakano and Professor T. Noda who have appreciated and used the author’s subloading surface model widely in their analyses and who have offered discussion continually on deformation of geomaterials. In addition, the author thanks Professor T. Nakai and Professor T.F. Zhang of the Nagoya Institute of Technology, for their valuable comments. The author is deeply indebted to Professor Bogdan Raniecki and H. Petryk of the Inst. Fund. Tech. Research, Poland, who have visited Kyushu University several times to deliver lectures on applied mechanics. Bogdan gave me valuable comments and suggestions by the critical reading of the manuscript, which highly

Preface

IX

contributed to the correction. Further, the author thanks Professor I.F. Collins of the University of Auckland, Professor O.T. Bruhns of Ruhr Univ., Bochum, Professor A.C. Aifantis of Michigan Tech. Univ. and Professor I. Vardoulakis of Natl. Univ. Tech. Athens, who have also stayed at Kyushu Univ., delivering lectures and engaging in valuable discussions related to continuum mechanics. The author wishes to acknowledge deeply Professor P. Wrrigers of Hanover Univ. who recommend me to publish this book in the series of Lecture Notes in Applied and Computational Mechanics, Springer. Finally, the author would like to state that the enthusiastic support of Mrs. Heather King from the Springer Publishing Company and SPS data processing team for their help in editing this book. Fukuoka December 2008

Koichi Hashiguchi

Contents Contents

1

Tensor Analysis ……………………………………………………… 1.1 Conventions and Symbols………………………………………... 1.1.1 Summation Convention…………………………………... 1.1.2 Kronecker’s Delta and Permutation Symbol……………... 1.1.3 Matrix and Determinant…………………………………... 1.2 Vector…………………………………………………………...... 1.2.1 Definition of Vector………………………………………. 1.2.2 Operations for Vectors……………………………………. 1.2.3 Component Description of Vector………………………... 1.3 Tensor……………………………………………………………. 1.3.1 Definition of Tensor………………………………………. 1.3.2 Quotient Law……………………………………………… 1.3.3 Notations of Tensors……………………………………… 1.3.4 Orthogonal Tensor………………………………………... 1.3.5 Tensor Product and Component…………………………... 1.4 Operations of Second-Order Tensor……………………………... 1.4.1 Trace……………………………………………………… 1.4.2 Various Tensors…………………………………………... 1.5 Eigenvalues and Eigenvectors………………………………….... 1.6 Calculations of Eigenvalues and Eigenvectors…………………... 1.6.1 Eigenvalues……………………………………………….. 1.6.2 Eigenvectors………………………………………………. 1.7 Eigenvalue and Eigenvectors of Skew-Symmetric Tensor………. 1.8 Cayley-Hamilton’s Theorem…………………………………….. 1.9 Positive Definite Tensor…………………………………………. 1.10 Polar Decomposition…………………………………………....... 1.11 Isotropic Tensor-Valued Tensor Function……………………….. 1.12 Representation of Tensor in Principal Space…………………….. 1.13 Two-Dimensional State………………….……………………….. 1.14 Partial Differential Calculi……………..……………………….... 1.15 Time Derivatives……………………..………………………..…. 1.16 Differentiation and Integration in Field……………………..……

1 1 1 1 2 8 8 8 9 14 14 15 17 18 19 20 20 21 26 31 31 32 33 35 35 36 37 40 44 47 50 51

2

Motion and Strain (Rate)……………………………………………. 2.1 Motion and Deformation…………………………………………. 2.1.1 Material, Spatial and Relative Descriptions………………. 2.1.2 Deformation Gradient and Deformation Tensors…………

57 57 57 59

XII

Contents

2.2 Strain Tensor……………………………………………………... 2.3 Strain Rate and Spin Tensors…………………………………….. 2.4 Various Simple Deformations……………………………………. 2.4.1 Uniaxial Loading………………………………………….. 2.4.2 Simple Shear……………………………………………… 2.4.3 Combination of Tension and Distortion…………………... 2.5 Surface Element, Volume Element and Their Rates……………...

64 70 81 81 84 94 97

3

Conservation Laws and Stress Tensors………………………..…… 3.1 Conservation Law of Mass………………………………………. 3.2 Conservation Law of Momentum………………………………... 3.3 Conservation Law of Angular Momentum………………………. 3.4 Stress Tensor……………………………………………………... 3.5 Equilibrium Equation…………………………………………….. 3.6 Equilibrium Equation of Moment………………………………... 3.7 Virtual Work Principle……………………………………………

101 101 101 102 102 105 107 108

4

Objectivity and Corotational Rate Tensor…………………………. 4.1 Objectivity………………………………………………………... 4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities………………………………………………………… 4.3 Rate of State Variable and Corotational Rate Tensor……………. 4.4 Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative……………………………………. 4.5 Various Objective Stress Rate Tensors…………………………... 4.6 Work Conjugacy………………………………………………….

111 111

5

Elastic Constitutive Equations…….………………………………… 5.1 Hyperelasticity…………………………………………………… 5.2 Cauchy Elasticity………………………………………………… 5.3 Hypoelasticity…………………………………………………….

127 127 130 131

6

Basic Formulations for Elastoplastic Constitutive Equations…….. 6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate………………………….. 6.2 Conventional Elastoplastic Constitutive Equations……………… 6.3 Loading Criterion……………….………………………………... 6.4 Associated Flow Rule……………………………………………. 6.4.1 Positivity of Second-Order Plastic Work Rate: Prager’s Interpretation ……………………………………………... 6.4.2 Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis………………………………………………… 6.4.3 Positivity of Second-Order Plastic Relaxation Work Rate..……………………………………………………… 6.4.4 Comparison of Interpretations for Associated Flow Rule...

135

112 114 119 122 124

135 142 148 151 151 151 152 153

Contents

6.5 Anisotropy………………………………………………………... 6.5.1 Definition of Isotropy.…………………………………….. 6.5.2 Anisotropic Plastic Constitutive Equation………………... 6.6 Incorporation of Tangential-Inelastic Strain Rate………………... 6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory……………………………………………………….…… 7

8

9

10

Unconventional Elastoplasticity Model: Subloading Surface Model……………………………………….......................................... 7.1 Mechanical Requirements………………………………………... 7.1.1 Continuity Condition……………………………………… 7.1.2 Smoothness Condition……………………………………. 7.2 Subloading Surface Model……………………………………….. 7.3 Salient Features of Subloading Surface Model…………………... 7.4 On Bounding Surface and Bounding Surface Model…………….. 7.5 Incorporation of Anisotropy……………………………………... 7.6 Incorporation of Tangential Inelastic Strain Rate………………...

XIII

156 156 157 159 165

171 171 171 172 174 181 184 186 187

Cyclic Plasticity Model: Extended Subloading Surface Model…… 8.1 Classification of Cyclic Plasticity Models……………………….. 8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening………………………..………...…………. 8.2.1 Multi-surface Model……………………………………… 8.2.2 Two-Surface Model………………………………………. 8.2.3 Infinite-Surface Model……………………………………. 8.2.4 Nonlinear Kinematic Hardening Model…………………... 8.3 Extended Subloading Surface Model…………………………….. 8.4 Modification of Reloading Curve………………………………... 8.5 Incorporation of Tangential-Inelastic Strain Rate………………...

191 191

Viscoplastic Constitutive Equations………………………………… 9.1 History of Viscoplastic Constitutive Equations………………….. 9.2 Mechanical Response of Ordinary Overstress Model……………. 9.3 Modification of Overstress Model: Extension to General Rate of Deformation…………………………………………………… 9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model…………………………………………………

211 211 214

Constitutive Equations of Metals………………………………….... 10.1 Isotropic and Kinematic Hardening…………………………….. 10.2 Cyclic Stagnation of Isotropic Hardening……………………… 10.3 On Calculation of the Normal-Yield Ratio……………………... 10.4 Comparisons of Test Results…………………………………… 10.5 Orthotropic Anisotropy………………………………………….

191 191 194 195 195 196 205 208

215 217 221 221 225 232 232 238

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Contents

10.6 Representation of Isotropic Mises Yield Condition……………. 10.6.1 Plane Stress State……………………………………….. 10.6.2 Plane Strain State………………………………………..

244 245 248

Constitutive Equations of Soils……………………………………… 11.1 Isotropic Consolidation Characteristics………………………… 11.2 Yield Conditions………………………………………………... 11.3 Isotropic Hardening Function…………………………………... 11.4 Rotational Hardening…………………………………………… 11.5 Extended Subloading Surface Model…………………………... 11.6 Partial Derivatives of Subloading Surface Function……………. 11.7 Calculation of Normal-Yield Ratio……………………………... 11.8 Simulations of Test Results…………………………………….. 11.9 Simple Subloading Surface Model……………………………... 11.10 Super-Yield Surface for Structured Soils in Natural Deposits……………………………………………………….. 11.11 Numerical Analysis of Footing Settlement Problem…………..

249 249 253 259 261 265 267 271 275 281

12

Corotational Rate Tensor…………….…………….…………….….. 12.1 Hypoelasticity…………….…………….…………….………… 12.1.1 Jaumann Rate…………….…………….………………. 12.1.2 Green-Naghdi Rate…………….…………….………… 12.2 Kinematic Hardening Material…………….…………………… 12.2.1 Jaumann Rate…………….…………….………………. 12.2.2 Green-Naghdi Rate…………….…………….………… 12.3 Plastic Spin…………….…………….…………….…………....

309 309 309 311 313 315 316 317

13

Localization of Deformation………………………………………… 13.1 Element Test…………….…………….…………….………….. 13.2 Gradient Theory…………….…………….…………….………. 13.3 Shear-Band Embedded Model: Smeared Crack Model………… 13.4 Necessary Condition for Shear Band Inception…………………

327 327 328 331 333

14

Numerical Calculation………………………………………………. 14.1 Numerical Ability of Subloading Surface Model………………. 14.2 Return-Mapping Algorithm Formulation for Subloading Surface Model…………………………………………………..

337 337

Constitutive Equation for Friction………………………………….. 15.1 History of Constitutive Equation for Friction…………………... 15.2 Decomposition of Sliding Velocity………………….................. 15.3 Normal Sliding-Yield and Sliding-Subloading Surfaces……….. 15.4 Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio………..…………………............................. 15.4.1 Evolution Rule of Sliding-Hardening Function……….. 15.4.2 Evolution Rule of Normal Sliding-Yield Ratio………...

349 349 350 354

11

15

291 301

340

355 355 356

Contents

15.5 15.6 15.7 15.8

XV

Relations of Contact Traction Rate and Sliding Velocity……… Loading Criterion………………….............................................. Sliding-Yield Surfaces…………………....................................... Basic Mechanical Behavior of Subloading-Friction Model…….. 15.8.1 Relation of Tangential Contact Traction Rate and Sliding Velocity…………………...…………………..... 15.8.2 Numerical Experiments and Comparisons with Test Data…………………....................................................... 15.9 Extension to Orthotropic Anisotropy…………………................

357 359 360 365

Appendixes……………………………………………………………. Appendix 1: Projection of Area………………………………………... Appendix 2: Proof of ∂ ( F jA / J ) / ∂ x j = 0 ................................................ Appendix 3: Euler’s Theorem for Homogeneous Function…………… Appendix 4: Normal Vector of Surface………………………………... Appendix 5: Convexity of Two-Dimensional Curve………………….. Appendix 6: Derivation of Eq. (11.19) ……………………………….. Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State………………………………………………...

387 387 388 388 389 390 391

References……………………………………………………………... Index……………………………………………………………………

395 407

366 367 375

392

Chapter 1

Tensor Analysis 1 Tensor Analysis Physical quantities appearing in deformation mechanics of solids belong mathematically to tensors. Therefore, their relations are described using tensor equations. Before studying the main theme of this book, elastoplaticity theory, the mathematical properties of tensors and the mathematical rules on tensor operations are explained to the extent necessary to understand elasoplasticity theory. The orthogonal Cartesian coordinate system is adopted throughout this book.

1.1 Conventions and Symbols Some basic conventions and symbols appearing in the tensor analysis are described in this section.

1.1.1 Summation Convention We first introduce the Cartesian summation convention. A repeated suffix in any term is summed over numbers that the suffix can take. For instance, n n ⎫ ur vr = ∑ ur vr , Tir vr = ∑ Tir vr ⎪ ⎪ r =1 r =1 ⎬ n ⎪ Trr = ∑ Trr ⎪⎭ r =1

where the range of suffixes is 1, 2,

• ••,

n .

(1.1)

Because of ur vr = us vs ,

Tir vr = Tis vs , Trr = Tss a letter of the repeated suffix is arbitrary. It is therefore called

as the dummy index. The convention described above is also called Einstein’s summation convention. Hereinafter, we assume that a repeated suffix obeys this convention, except for the particular case in which it is stated that this convention is not applied.

1.1.2 Kronecker’s Delta and Permutation Symbol The symbol δ ιj (i , j = 1, 2, …, n) defined in the following equation is called the Kronecker’s delta. K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 1–56. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

2

1 Tensor Analysis

⎧1: i = j 　 ⎩0 : i ≠ j

δ ij = ⎨

(1.2)

for which it holds that

δir δrj = δij ,

δii = n

(1.3)

Furthermore, the symbol ε p1 p2…pn defined by the following equation is called the alternating (or permutation) symbol or Eddington’s epsilon or Levi-Citiva “e” tensor.

⎧ l ε p1 p2…pn = ⎨⎧⎨(−1) : p1 p2 ... pn 　are different from each other (1.4) ⎩ 0 : others

where l is the number of replacement required to obtain the permutation p1 , p2 , ..., pn from the regular permutation 1, 2, ..., n . Equation (1.4) is written for the third degree (n = 3) as

⎧ 1 : for ijk is a cyclic permutation of 123 ε ijk = ⎪⎨−1 : for ijk is an anticyclic permutation of 123 ⎪ 0 : others ⎩

(1.5)

The number of permutations that the suffixes p1 , p2 , ..., pn in ε p1 p2…pn take different values from each other is n! and, needless to say, the square of

ε p1 p2…pn (=1 or

− 1) is +1 . Therefore, it holds that

ε p1 p2… pn ε p1 p2…pn = n!

(1.6)

1.1.3 Matrix and Determinant When the quantity T possessing n × n components Tij is expressed in the arrangement

⎡T11 T12 …T 　 1n ⎤ ⎢T T …T ⎥ 2n ⎥ ⎢ 21 22 T = [Tij ] = ⎢ ⎥ ⎢ ⎥ ⎢⎣Tn1 Tn 2 …Tnn ⎥⎦

(1.7)

the expression of T in this form is called a matrix. For the two matrices T and S , their product TS is defined by the matrix having the following components.

1.1 Conventions and Symbols

3

(1.8)

(TS)ij = Tir Srj

Further, the quantity defined by the following equation is called the determinant of T and is shown by the symbol det T , i.e.

det T = ε p1 p2… pn T1 p1T2 p2 …Tn pn

(1.9)

which for n = 3 becomes

det T = ε pqr T1 pT2 qT3 r

(1.10)

Therefore, the determinant is the number obtained using the process by which i) one makes product by selecting elements in different rows from all lines (number of products made in this process is n ! ) and ii) sums them up with signs which are plus for even permutation and minus for odd permutation, i.e. multiplying the permutation symbol. Now, consider the rearrangement of the elements

T1 p1T2 p2 …Tn pn → Tr11Tr2 2 …Trn n changing the order such that the second suffices becomes the regular permutation, while the first suffices was the regular order before the rearrangement. Here, the numbers of times in the replacement to obtain the permutations p1 , p2 , , pn and r1, r2 , , rn from the regular permutation 1, 2, ..., n are the same. Then, Eq. (1.9) can be rewritten as follows:

…

…

det T = ε r1r2…rn Tr11Tr2 2 …Trn n

(1.11)

Eventually, “the value of determinant does not change even if the lines and the rows are mutually substituted”, i.e.

det TT = det T

(1.12)

where ( )T designates the transpose, i.e. the mutual replacement of lines and rows. Here, the number of permutations that the suffixes p1 , p2 , ..., pn in

ε p1 p2…pn can

take is n! . Therefore, Eqs. (1.9) and (1.11) can be written collectively as

det T = 1 ε r1r2…rn ε p1 p2… pn Tr1 p1Tr2 p2 …Trn pn n!

(1.13)

which is written for the third degree matrix as

det T = 1 ε abcε pqrTapTbqTcr 3!

(1.14)

4

1 Tensor Analysis

Eq. (1.13) is rewritten as

det T = n1 Trs Δrs , det T = 1n TΔT

(1.15)

where

Δij ≡ 1 ε i r1r2…rn−1ε j p1 p2… p n−1 Tr1 p1Tr2 p2 …Trn−1 pn −1 (1.16) (n − 1) ! which is called the cofactor for the i -line and the j -row. Equations. (1.13) and (1.16) are written for the third degree matrix as

det T = 1 Trs Δ rs, Δij ≡ 1 ε rabε s pq Tab Tpq 3 2 Because of

(1.17)

ε p1 p2… pi… p j … pn = − ε p1 p2…p j …pi…pn meaning the change of

sign and T1 p1T2 p 2

…Tipi …Tj p …Tn p j

n

= T1 p1T2 p2 … Tj p j …Tipi

…Tn p

n

it

follows that

ε p1 p2… pi… p j … pnT1 p1T2 p2 … Ti pi … Tj p j … Tn pn

= −ε p1 p2… p j … pi… pn T1 p1T2 p2 …Tj p j …Ti pi …Tn pn

(1.18)

Then, “the sign of the determinant changes if two lines or two rows ( i -line and j - row in the above equation) are mutually substituted”. This property engenders the fact “the determinant having two lines or rows that are mutually same is zero”. Multiplying ε r1r2…rn to both sides in Eq. (1.9), one has

ε r1r2…rn detT = ε r1r2…rn ε p1 p2…pn T1 p1T2 p2 …Tn pn

= ε p1 p2…pn Tr1 p1Tr2 p2 …Trn pn

(1.19)

The transformation from the second side to the third side in Eq. (1.19) results from rn , as the fact that the determinant is zero if the same number exists in r1r2 rn respectively described in Lemma (1.18), and is detT and − detT if r1r2 signify the even permutation and odd permutation. Here, note that the expression of the determinant in Eq. (1.13) is obtained by multiplying ε r1r2…rn to both sides in Eq. (1.19) and noting Eq. (1.6).

…

…

The additive decomposition of the elements Ti p j in the determinant

T1 p1 T2 p2 …Ti p j …Tn pn into Ti pj = Ai pj + Bi pj leads to

ε p1 p2…p j…pn

ε p1 p2…p j…pn T1 p1T2 p2 …( Ai pj + Bi pj )…Tn pn

= ε p1 p2 … p j … pn T1 p1 T2 p2 … Ai p j …Tn pn

1.1 Conventions and Symbols

5

+ ε p1 p2… p j … pnT1 p1T2 p2 …Bi pj …Tn pn

(1.20)

Therefore, the value of determinant in which elements in a line (or row) are decomposed additively is the sum of the two determinants made by exchanging the line (or row) of the original determinants into the decomposed elements. Since it is obtained from Eqs. (1.8), (1.9), (1.11) and (1.19) that

ε p1 p2…pn (T1r1 Sr1 p1 )(T2 r2 Sr2 p2 )…(Tnrn Srn pn ) = ε p1 p2… pn T1r1T2 r2 …Tnrn Sr1 p1 Sr2 p2 … Srn pn = ε r1r2…rn T1r1T2 r2 …Tnrn det S

(1.21)

the following product law of determinant holds. (1.22)

det(TS) = detTdetS The partial derivative of a determinant is given from Eq. (1.13) as

∂det T = ∂ Tij

1 ∂ n! ε r1r2…rn ε p1 p2… pn Tr1 p1Tr2 p2 …Trn pn ∂ Tij

= 1 (ε r1r2…rn ε p p … pn δ i r1δ j p1Tr p …Trn pn 1 2 2 2 n! + ε r1r2…rn ε p p … pn Tr p δ i r2δ j p2 …Trn pn 1 2

+ •••

1 1

+ ε r1r2…rn ε p p … pn Tr p δ i r2δ j p2 …δ i rnδ j pn 1 2

1 1

= 1 (ε ir2…rn ε j p … pn Tr p …Trn pn 2 2 2 n! + ε r1i…rn ε p j… pn Tr p Tr3 p …Trn pn + •••

1

1 1

3

+ ε r1r2…iε p p … jTr p Tr p 1 2

1 1

2 2

…Tr − p − Tr p Tr p …Trn

n = (n − 1) ! ε i r1r2…r n−1ε j p1 p2… p n−1

n 1 n 1

1 1

2 2

−1 pn −1

which leads to

∂ det T = Δ , ∂ det T = Δ ij ∂ Tij ∂T

(1.23)

6

1 Tensor Analysis

Equation (1.23) is derived for the third degree matrix as follows:

1ε ε T T T ∂det T = ∂ 3! abc pqr ap bq cr ∂ Tij ∂ Tij 1 = 3! ε abcε pqr (δ iaδ jpTbqTcr + Tapδ ibδ jqTcr + TapTbqδ icδ jr ) =

1 ε ε + ε aicε pj rTapTcr + ε abiε pqjTapTbq ) ( 3! ibc jqrTbqTcr

1 = 3! (ε ibcε jqrTbqTcr + ε bicε qjr TbqTcr + ε cbiε rqjTcrTbq ) 1 = 2! ε ibcε j qrTbqTcr = Δ ij The permutation symbol in the third order, i.e.

ε ijk

appears often hereinafter. It

is related to Kronecker’s delta by the determinants.

δ1i δ1j δ1k ε ijk = δ 2i δ 2 j δ 2 k

δ1i δ 2i δ 3i = δ1 j δ 2 j δ 3 j

δ 3i δ 3 j δ 3 k

δ1k δ 2 k δ 3k

(1.24)

Here, the second side in Eq. (1.24) is expanded as

δ1i δ1j δ1k ε ijk = δ 2i δ 2 j δ 2 k = δ1iδ 2 j δ 3 k + δ1kδ 2iδ 3 j + δ1j δ 2 kδ 3i δ 3i δ 3 j δ 3 k − δ1 kδ 2 j δ 3i − δ1iδ 2 kδ 3 j − δ1 j δ 2iδ 3 k We can confirm this relation by

ε123 = δ11δ 22δ 33 + δ13δ 21δ 32 + δ12δ 23δ 31 − δ13δ 22δ 31 − δ11δ 23δ 32 − δ12δ 21δ 33 = 1 ε 213 = δ12δ 21δ 33 + δ13δ 22δ 31 + δ11δ 23δ 32 − δ13δ 21δ 32 − δ12δ 23δ 31 − δ11δ 22δ 33 = −1 for instance. The third side in Eq. (1.24) could be confirmed as well.

1.1 Conventions and Symbols

7

The following relation holds from Eqs. (1.24) and (1.22).

ε ijk ε pqr

⎡ δ1i δ 2i δ 3i ⎤ ⎡ δ1 p δ1 q δ1r ⎤ δ1i δ 2i δ 3i δ1 p δ1q δ1r ⎥ ⎢ ⎥⎢ = δ1 j δ 2 j δ 3 j δ 2 p δ 2 q δ 2 r = ⎢ δ1 j δ 2 j δ 3 j ⎥ ⎢ δ 2 p δ 2 q δ 2 r ⎥ ⎢ δ δ δ ⎥⎢ δ p δ q δ ⎥ δ1k δ 2 k δ 3k δ 3 p δ 3q δ 3 r ⎣ 1k 2 k 3k ⎦ ⎣⎢ 3 3 3 r ⎦⎥

δ siδ sp δ siδ sq δ s iδ sr

δ ip δ iq δ ir

= δ sjδ sp δ sjδ sq δ s jδ sr = δ jp δ jq δ jr

δ skδ sp δ skδ sq δ s kδ s r

(1.25)

δ kp δ kq δ kr

from which further one has

δ ip δ iq δ ik ε ijk ε pqk = δ jp δ jq δ jk δ kp δ kq δ kk = δ ipδ jqδ kk + δiqδ jk δ kp + δ ikδ jpδ kq − δikδ jqδ kp − δipδ jk δ kq − δiqδ jpδ kk = 3δ ipδ jq + δ iqδ j p + δ iqδ j p − δ ipδ jq − δ ipδ jq − 3δ iqδ jp = δ ipδ jq − δ iqδ jp

δ ii δ ij δ iq ε ijpε ijq = δ ji δ jj δ j q δ pi δ pj δ pq = δ iiδ jj δ pq + δ ijδ j qδ pi + δ iqδ jiδ pj − δ iiδ j qδ pj − δ ijδ jiδ pq − δ iqδ jj δ pi = 9δ pq + δ iqδ pi + δ iqδ ip − 3δ pq − 3δ pq − 3δ pq = 2δ pq ε ijk ε ijk = 2δ kk = 6 Consequently, the following relation holds.

ε ijk ε pqk = ε kij ε kpq = δ ipδ jq − δiq δ jp ⎫⎪ ε ijpε ijq = 2δ pq , ε ijk ε ijk = 6

⎬ ⎪⎭

The last equation can also be obtained directly from Eq. (1.6).

(1.26)

8

1 Tensor Analysis

1.2 Vector 1.2.1 Definition of Vector A quantity having only magnitude is defined as a scalar. On the other hand, a quantity having direction and sense in addition to magnitude and fulfilling the following three properties is defined as a vector. A vector is expressed using lowercase letters in boldface to distinguish it from a scalar. Equivalence: The vectors having same magnitude, direction and sense are equivalent. Here, equivalence of two vectors u and v is expressed by u = v . Addition: The addition of vectors is given by the parallelogram law. Multiplication with scalar: The multiplication of vector and scalar induces a vector whose magnitude is given by the multiplication of the original vector and the scalar, direction is identical to that of the original vector, and sense is same and opposite to that of the original vector if the scalar is positive and negative, respectively. By virtue of the properties presented above, the commutative, distributive, and the associative laws hold as follows:

u + v = v + u, 　　 (u + v) + w = u + ( v + w )

⎫ ⎬ a(bv) = (ab) v = b(av), (a + b)v = (b + a)v, a(u + v) = au + av ⎭ (1.27)

where a, b are arbitrary scalars. The magnitude of vector is denoted by || v || . In particular, the vector whose magnitude zero is called the zero vector and is shown as 0 . The vector whose magnitude is unity is called the unit vector.

1.2.2 Operations for Vectors 1) Scalar product Denoting the angle between the two vectors u, v by θ when they are translated to the common initial point, the scalar (or inner) product is defined as ||u |||| v ||cosθ and it is denoted by the symbol u • v , i.e.

u • v ≡ ||u |||| v || cosθ

(1.28)

The magnitude of vector is expressed using the scalar product in Eq. (1.28) as follows:

|| v ||= v • v

(1.29)

The quantity obtained by the scalar product is a scalar and the following commutative, distributive and associative laws hold.

u • v = v • u, v • 　 (u + w) = u • v + u • w, a(u • v) = (au) • v (1.30)

1.2 Vector

9

2) Vector product The operation obtaining a vector having 1) magnitude identical to the area of the parallelogram formed by the two vectors u, v , provided that they are translated to the common initial point, and 2) direction of the unit vector n which forms the right-hand bases u, v, n in this order is defined as the vector (or cross) product and is noted by the symbol u × v . Therefore, denoting the angle between the two vectors u, v by θ when they are translated to the common initial point, it holds that

u × v ≡ || u |||| v || sinșn ( || n|| = 1)

(1.31)

The vector product is not commutative, i.e.

u × v= − v × u

(1.32)

On the other hand, the distributive and the associative laws hold as follows:

u ×( v + w) = u × v + u × w,

(au) × (bv) = ab(u × v)

(1.33)

The operation defined by the following equation for the vector and the scalar products of three vectors is called scalar triple product.

[uvw ] ≡ (u × v ) • w

(1.34)

The commutative law for the scalar triple product will be shown in the subsequent section. In addition to the scalar and the vector products, the tensor product is defined as will be described in 1.3.5.

1.2.3 Component Description of Vector The component description of vector is explained here prior to the description of component description of general tensor.

1) Component description Consider the set of the normalized orthonormal vectors {ei } (i = 1, 2,…, n) . Here, the “normalized” means the unit vector and “orthonormal” means that they are mutually orthonormal. Then, adopt the coordinate system {O − x i} , in which the directions of axes are chosen to the directions of normalized orthonormal vectors {ei } . Hereinafter, the set {ei } of vectors is called the normalized orthonormal base. The scalar and the vector products between the base vectors for n = 3 are given from Eq. (1.2), (1.28) and (1.31) as follows:

ei • e j = δ ij

(1.35)

10

1 Tensor Analysis

ei × ej = ε ijr er Vector

(1.36)

v is described in the linear associative form as follows: v = vr er (=v1e1 + v2e2 + … + vnen )

(1.37)

where v1 , v2 , …, vn are the components of v . Denoting the angle of the direction of vector v from the direction of the base vector e i by θ i , cosθ i = n • ei is called

the direction cosine by which the component of v is given as

vi = v • ei = || v || n • ei = || v ||cosθ i

(1.38)

The magnitude of vector v and its unit direction vector n are given from Eqs. (1.29), (1.35), (1.37) as follows:

|| v || ≡ vr vr ,

v n ≡ v = r er || v || || v ||

(1.39)

Because of u • v = u r e r • vs es = ur vsδ rs the scalar product is expressed using the components as

u • v = ur vr

(1.40)

The vector product is expressed from Eqs. (1.36), (1.37) as follows:

u × v =ui ei × v j e j = ε ijk u j vk e i (= (u2 v3 − u3v2 )e1 + (u3v1 − u1v3 )e2 + (u1v2 − u2v1 )e3 ) (1.41) For sake of Eq. (1.41) the scalar triple product defined in Eq. (1.34) is expressed in a component form as

[uvw ] = (u × v ) • w = ε ijr ui v j e r • w k e k = ε ijr ui v j w k δ rk = ε ijk ui v j w k (1.42) or in a matrix form as follows:

u1 v1 w1 [uvw ] = u2 v2 w2

(1.43)

u3 v3 w3 Furthermore, from these equations the following equation holds.

[uvw ] = [ vwu] = [wuv] = −[ vuw ] = −[wvu] = −[uwv]

(1.44)

1.2 Vector

11

Here, note that [uvw] designates the volume of a parallelepiped formed by choosing u, v, w as the three sides to produce the right-handed coordinates in this order. 2) Coordinate transformation Adopt the other normalized orthogonal coordinate system {O - xi∗} with the bases

{e*i } in addition to the normalized orthogonal coordinate system {O − x i} with the bases {ei } (Fig. 1.1). Noting v = v j e j = ( v • e j )e j in general, the following relations hold in these bases. ei = (ei • e*j )e*j ⎫⎪ ⎬ e*i = (e*i • e j )e j ⎪ ⎭

(1.45)

ei = Qri e*r ⎪⎫ ⎬ e*i = Qir e r ⎪⎭

(1.46)

where the coordinate transformation operator Qij is defined by

Qij ≡ cos(angle between e*i and e j ) = e*i • e j x2 x *2

v

v2

v*2

e2 v*1

e*2

e*1 0

e1

x*1

θ v1

x1

Fig. 1.1 Coordinate transformation of vector in a two-dimensional state

(1.47)

12

1 Tensor Analysis

Moreover, because of

Qir Q jr = (e*i • e r )(e*j • e r ) = e*i • (e*j • e r )e r = e*i • e*j ⎫⎪ ⎬ QriQrj = (e*r • ei )(e*r • e j ) = (ei • e*r )e*r • e j = ei • e j ⎪ ⎭ it holds that

Qir Q jr = QriQrj = δij

(1.48)

It is assumed for a while that the relative (parallel and rotational) motion does not exist between the above-described coordinate systems, and that their origins mutually coincide. Then, denoting the component on the base e*i by ( )* , the coordinate transformation rule, i.e. the transformation rule of the components of v on these coordinate systems is given by

vi∗ = Qij v j

(1.49)

noting

v*r e*r • e*i = vj e j • e*i and based on

v = v j e j = v*r e*r

(1.50)

Furthermore, noting Qri v∗ r = Qri Qrs vs = δ is vs, the inverse relation of Eq. (1.49) is given as

vi = Qji v∗j

(1.51)

Equations (1.49) and (1.51) are expressed in matrix form as

⎧v* ⎫ ⎡Q Q Q ⎤ ⎪ 1 ⎪ ⎢ 11 12 13 ⎥ ⎧v1 ⎫ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨v*2 ⎬ = Q21 Q22 Q23 ⎥ ⎨v2 ⎬ , ⎥⎪ ⎪ ⎪ ⎪ ⎢ v ⎢ * Q Q Q v ⎪⎩ 3 ⎪⎭ ⎣ 31 32 33 ⎥⎦ ⎩ 3 ⎭

⎡ ⎤ ⎧ *⎫ ⎧v1 ⎫ ⎢Q11 Q21 Q31 ⎥ ⎪v1 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨v2 ⎬ = Q12 Q22 Q32 ⎥ ⎨v*2 ⎬ ⎥⎪ ⎪ ⎪v ⎪ ⎢ ⎩ 3 ⎭ ⎢Q Q Q ⎥ ⎪v3* ⎪ ⎣ 13 23 33 ⎦ ⎩ ⎭

(1.52)

which are often expressed simply as

{v*} = [Q]{v}, {v} = [Q]T {v*}

(1.53)

1.2 Vector

13

Needless to say, an equation involving ( )* or Qi j does not describe the relation between different physical quantities, but describes the relations between components when a certain physical quantity is described by two different coordinate systems. As known from the following equation, the vector of magnitude is not influenced by the coordinate transformation, whilst it is the basic property of the scalar quantity, as described later.

||v *|| = Qir vr Qis vs = Qir Qis vr vs = δ rs vr vs = vr vr = ||v|| The relations expressed above are shown below.

⎡e* • e e* • e ⎤ ⎡ (e* ) (e* ) ⎤ ⎡ cosθ sinθ ⎤ 1 2 ⎥=⎢ [Q ] = ⎢ 1 1 1 2 ⎥ = ⎢ 1 1 ⎥ ⎢e* • e e* • e ⎥ ⎢ (e* ) (e* ) ⎥ ⎣ _sinθ cosθ ⎦ ⎣ 2 1 2 2 ⎦ ⎣ 2 1 2 2⎦

cos θ sinθ ⎤ ⎡cos θ _sinθ ⎤ ⎡1 0 ⎤ Qir Q jr = ⎡⎢ ⎥⎢ ⎥ = ⎢ ⎥ = ⎡⎣δ ij ⎤⎦ _ ⎣ sinθ cos θ ⎦ ⎣ sinθ cos θ ⎦ ⎣0 1⎦

e∗i = (e∗i • e1 )e1 + (e∗i • e2 )e2 = Qi1e1 + Qi 2e2 , e1∗ = cos θ e1 + sin θ e2 ⎪⎫ ⎬ e∗2 = − sin θ e1 + cos θ e2 ⎪⎭ ei = (ei • e1∗ )e1∗ + (ei • e∗2 )e∗2 = Q1i e1∗ + Q2 i e∗2

v = v1e1 + v2e 2 = v1∗e1∗ + v∗2 e∗2 ⎧⎪v1∗ ⎫⎪ ⎡ cos θ sinθ ⎤ ⎧v1 ⎫ ⎧v ⎫ ⎡cos θ _sinθ ⎤ ⎧v* ⎫ ⎪1⎪ 1 ⎨ ⎬ = ⎢_ ⎥ ⎨v ⎬ , ⎨ ⎬ = ⎢ ⎥ ⎨ v∗ ⎬ * sin θ cos θ v v sin cos θ θ ⎦⎩ 2⎭ ⎩ 2⎭ ⎣ ⎦ ⎩⎪ 2 ⎭⎪ ⎪⎩ 2 ⎪⎭ ⎣ Choosing the position vector x as the vector v , it holds from Eqs. (1.49) and (1.51)that

x∗i = Qir xr ⎫⎪ ⎬ xi = Q ri x∗r ⎪⎭

(1.54)

14

1 Tensor Analysis

from which one has

⎫ ∂xi* ∂Qir xr Q ∂xr = = ir = Qirδ jr = Qij ⎪ ∂x j ∂x j ∂x j ⎪ ⎬ ∂x j ∂Qrj x∗r Q ∂ x∗r ⎪ = = rj ∗ = Qrjδ ir = Qij ⎪ ∂ xi ∂xi* ∂xi* ⎭

(1.55)

Consequently, Qij can be also described as

Qij =

∂ xi∗ ∂x j = ∂x j ∂ x∗i

(1.56)

1.3 Tensor The vector described in the foregoing possesses the direction in first order but there exit quantities possessing the direction in high order. They are collectively called the tensor. The general definition and mathematical properties of tensor are described in this section.

1.3.1 Definition of Tensor When the set of n m functions T ( p1 , p2 , ..., pm ) , where each of the suffixes p1 , p2 , ..., pm takes the number 1, 2, , n , is described in the coordinate system {O - xi } , this set of functions is defined as the mth-order tensor in the nth - dimension if the set of functions is observed as

T *( p1 , p2 ,

pm ) = Qp1q1 Qp2q2

Qpmqm T (q1 , q2 ,

qm )

(1.57)

or

T * ( p1 , p2 ,

pm ) =

∂x∗p1 ∂x∗p2 ∂xq1 ∂xq2

∂x ∗pm T ( q1 , q2 , ∂xqm

qm )

(1.58)

in the other coordinate system, provided that only the directions of axes are different but the origin is common and the relative motion does not exist. It is denoted by the symbol Tp1 p2⋅⋅⋅ pm . Then, Eq. (1.58) is expressed as

Tp∗1 p2 ⋅⋅⋅ pm = Q p1q1 Q p q … Q pm qm Tq1q ⋅⋅⋅qm 2 2 2

(1.59)

1.3 Tensor

15

or

Tp∗1 p2⋅⋅⋅pm =

∂x∗p1 ∂x∗p2 ∂xq1 ∂xq2

∂x∗pm T ∂xqm q1q2⋅⋅⋅qm

(1.60)

Noting that

Q p1r1 Q p2r2

Q pmrm Tp∗1 p2⋅⋅⋅ pm = Q p1r1 Q p2r2

Q pm rm Q p1q1 Q p2q2

= (Q p1r1 Q p1q1 )(Q p2r2 Q p2q2 ) = δ r1q1δ r2q2

• • •

•• •

Qpm qm Tq1q2 ⋅⋅⋅qm

(Q p r Q p q )Tq1q2 ⋅⋅⋅qm m m m m

δ r q Tq q ⋅⋅⋅q m m

1 2

m

the inverse relation of Eq. (1.59) is given by

Tr1r2⋅⋅⋅rm = Q p1r1 Q p2r2

Q pm rm Tp∗1 p2 ⋅⋅⋅ pm

(1.61)

While the transformation rule of the first-order tensor, i.e. vector is given by Eqs. (1.49) and (1.51), the transformation rule of the second-order tensor is given by

Tij∗ = Qir Q jsTrs , Tij = Qri Q jsTrs∗

(1.62)

The transformation between the coordinate systems without relative motion is considered above in the definition of the tensor, whereas the transformation in the form of (1.59) or (1.61) is called as the objective transformation. A tensor that obeys the objective transformation even between the coordinate systems with the relative motion is called an objective tensor.

1.3.2 Quotient Law One has a convenient law, called the quotient law, which is used to judge whether or not a quantity is a tensor. It will be explained below.

Quotient law: “If a set of functions T ( p1 , p2 , • • •, pm ) becomes B pl +1 pl +2⋅⋅⋅ pm ( m − l - th order tensor lacking the suffices p1 ∼ pl ) by multiplying it by A p1 p2⋅⋅⋅ pl ( l - th order tensor (l ≤ m) ), the set is a m - th order tensor”. (Proof) The proof is achieved by showing that the quantity T ( p1 , p2 , • • •, pm ) is the m - th order tensor when it holds that T ( p1 , p2 , • • •, pm ) Ap1 p2⋅⋅⋅pl

= B pl+1 pl +2⋅⋅⋅ pm

(1.63)

16

1 Tensor Analysis

which is described in the coordinate system {O - xi∗ } as follows:

T ∗ ( p1 , p2 , • • •, pm ) A∗p1 p2 ⋅⋅⋅ pm = B*pl +1 pl +2 ⋅⋅⋅ pm

(1.64)

Here, A p1 p2⋅⋅⋅ pl is the l - th order tensor and B pl +1 pl +2⋅⋅⋅ pm is the m − l - th order tensor,. Therefore, the following relation holds.

B*pl +1 pl +2⋅⋅⋅ pm = Q pl +1rl +1 Q pl +2rl +2

= Qp

l +1rl +1

Q pl +2 rl +2

•••

= Q pl+1rl+1 Q pl+2rl+2

•••

Q pm rm Brl +1rl +2⋅⋅⋅rm

Q pm rm T ( r1 , r2 , • • •, rm ) Ar1r2 ⋅⋅⋅rl

•••

Q pmrm T (r1 , r2 , • • •, rm )Q p1r1 Q p2r2

(l + 1 ∼ m)

•••

Q pl rl A∗p1 p2⋅⋅⋅pl

(1.65)

(1 ∼ l )

Substituting Eq. (1.64) into Eq. (1.65) yields

{T ∗ ( p1 , p2 , • • •, pm ) − Q p1r1 Q p2 r2

•••

Q pm rm T (r1 , r2 , • • •, rm )} Ap1 p2⋅⋅⋅pl = 0 (1.66)

from which it holds that

T ∗ ( p1 , p2 , • • •, pm ) = Q p1r1 Q p2 r2

•••

Q pm rm T (r1 , r2 , • • •, rm ) (1.67)

Therefore, noting the definition of tensor in Eq. (1.67), the quantity (End of proof) T ( p1 , p2 , • • •, pm ) is the m - th order tensor. According to the proof presented above, Eq. (1.63) can be written as

Tp1 p2⋅⋅⋅pm Ap1 p2⋅⋅⋅pl = B pl +1 pl +2⋅⋅⋅pm

(1.68)

For instance, if the quantity T (i, j ) transforms the first-order tensor, i.e. vector v i to the vector ui by the operation T (i, j )v j =u i , one can regard T (i, j ) as the second-order tensor. Eventually, in order to prove that a certain quantity is a tensor, one needs only to show that it obeys the tensor transformation rule (1.59) or that the multiplication of a tensor to the quantity leads to a tensor by the quotient rule. Tensors fulfill linearity as follows:

Tp1 p2⋅⋅⋅pm (G p1 p2⋅⋅⋅pl + H p1 p2⋅⋅⋅pl ) = Tp1 p2⋅⋅⋅pm G p1 p2⋅⋅⋅ pl + Tp1 p2⋅⋅⋅pm H p1 p2⋅⋅⋅pl ⎫⎪ ⎬ Tp1 p2⋅⋅⋅pm (aAp1 p2⋅⋅⋅pl ) = aTp1 p2⋅⋅⋅ pm Ap1 p2⋅⋅⋅pl ⎪⎭ where a is an arbitrary scalar. Therefore, the tensor has the function to transform linearly a tensor to the other tensor and thus it is called the linear transformation. The operation that lowers the order of tensor by multiplying the other tensor is called the contraction.

1.3 Tensor

17

1.3.3 Notations of Tensors When we express the tensor T as

T = Tp1 p2⋅⋅⋅ pm e p1⊗e p2 • • • ⊗ e pm

(1.69)

in a similar form to the case of vector in Eq. (1.37), Eq. (1.69) is called the component notation with bases, defining e p1⊗e p2 • • • ⊗ e pm as the base of m - th order tensor. The transformation of T between the bases in Eq. (1.46) leads Eq. (1.69) to

T = Tp1 p2 ••• pm Qr1 p1 e∗r1 ⊗ Qr2 p2 er∗2 ⊗ • • • ⊗ Qrm p m e∗rm

= Qr p Qr 1 1

p •••

2 2

Qrm pm Tp1 p2⋅⋅⋅pm e∗r1 ⊗ e∗r2 ⊗ • • • ⊗ e∗rm

= Tr1∗r2 ••• rm e∗r1 ⊗ e∗r2 ⊗ • • • ⊗ e∗rm

(1.70)

The following various notations are used for tensors.

Component notation: T p1 p2 ⋅⋅⋅ pm Component notation with base: T p1 p2 ⋅⋅⋅ pm e p1 e p2 ⊗

⋅⋅⋅ ⊗ e

pm

Symbolic (or direct) notation: T Matrix notation: Eq. (1.7) The matrix notation holds only for a vector or a second-order tensor or for a fourth-order tensor if it is formally expressed by two suffixes. For instance, the stress-strain relation can be expressed in matrix notation by expressing the stress and the strain of second-order tensors as a form of vector and the stiffness coefficient of fourth-order tensor as a form of second-order tensor. Various contractions exist in the operation of higher-order tensors and thus the symbolic notation is not useful in general. For instance, which of the following does ST mean: Sijk Tjk , Sijk Tkj , Sijk Tij , Sijk Tji , Sijk Tj l , Sijk Tkl , Sijk Til ? In other words, symbolic notation has its limit. On the other hand, component notation with bases holds always without setting any special rule. Introducing the notation

(Q T ) p p p ≡ Q p q Q p q • • • Q p q Tq q q ⎫⎪ m m 1 2 ⋅⋅⋅ m 1 1 1 2 ⋅⋅⋅ m 2 2 ⎬ (QT T ) p p p ≡ Q p q Q p q • • • Q p q Tq1q2⋅⋅⋅qm ⎪ m m 1 2 ⋅⋅⋅ m 1 1 2 2 ⎭

(1.71)

18

1 Tensor Analysis

Eqs. (1.59) and (1.61) can be expressed by the symbolic notation as follows: ⎫⎪ ⎬ T∗ ⎪⎭

T* = Q T T=

QT

(1.72)

In particular, transformations of the vector and the second-order tensor are expressed by

v ∗ = Qv, v = QT v ∗ T* = QTQ T , T = Q T T*Q

(1.73) (1.74)

1.3.4 Orthogonal Tensor The coordinate transformation operator Q i j described in 1.2.3 plays an important role in the coordinate transformation and is called the orthogonal tensor. The component notation with bases is obtained from

Qij ei ⊗ e j = ei ⊗ (e∗i • e j )e j ( = ei ⊗ e∗i ) = (ei • e∗r )e∗r ⊗ (e∗i • e j )e j

as follows:

= Qrie∗r ⊗ e∗i

(1.75)

Q = Qij ei ⊗ e j = Qije∗i ⊗ e∗j

(1.76)

Furthermore, considering Eq. (1.46), the direct notation of Q is given by

Q = ei ⊗ e∗i

(1.77)

Because of

ei = δ ir e r = Qis Qrs er = Qrs er Qis = Qrs e r (e s • e∗i ) = (Qrs er ⊗ e s )e∗i ⎫⎪ ⎬ e∗i = δ ir e∗r = Qsi Qsr e∗r = Qsr e∗r Qsi = Qsr e∗r (e∗s • ei ) = (Qsr e∗r ⊗ e∗s )ei ⎭⎪ it holds that

ei = Qe∗i , e∗i = QT ei Furthermore, changing Eq. (1.48) to the direct notation or noting the relation

QQT = ei ⊗ e∗i e∗j ⊗ e j = ei (e∗i • e∗j ) ⊗ e j = eiδ ij ⊗ e j ⎫⎪ ⎬ QT Q = e∗i ⊗ ei e j ⊗ e∗j = e∗i (ei • e j ) ⊗ e∗j = e∗i δ ij ⊗ e∗j ⎪ ⎭

(1.78)

1.3 Tensor

19

obtained from Eq. (1.77), it holds that QQT = QT Q = I

(1.79)

from which one has Q T = Q −1

(1.80)

Moreover, from Eqs. (1.12), (1.22) and (1.79), it is obtained that det Q = det Q T = ±1

(1.81)

Further from Eq. (1.79) one obtains (Q − I)QT = −(Q − I )T

Making the determinant of this equation and noting Eqs. (1.12), (1.22) and (1.81), it holds that

det(Q − I )= − det(Q − I ) → det(Q − I ) = 0

(1.82)

Then, it is known that one of the principal values of the orthogonal tensor is unity as known from the fact described in 1.5.1.

1.3.5 Tensor Product and Component Based on the three vectors v (1) , v (2) , ⋅⋅⋅, v ( m ) , one can make the m - th order tensor as follows: (2) ( m) v (1) ⊗ v (2) ⋅⋅⋅ ⊗ v ( m ) = v (1) p1 v p 2 ⋅⋅⋅ v p m e p1 ⊗e p2 ⋅⋅⋅⊗ e pm

(1.83)

For the two vectors, one has the second-order tensor

u ⊗ v = ui ei⊗v j e j = ui v j ei⊗e j

(1.84)

which is expressed in matrix form

⎡u1v1 u1v2 u1v3 ⎤ ⎢u v u v u v ⎥ ⎢ 2 1 2 2 2 3⎥ ⎢⎣u3v1 u3v2 u3v3 ⎥⎦

(1.85)

As described above, one can make a tensor from two vectors. After the scalar product u • v and the vector product u × v for the two vectors u , v , one calls u ⊗ v as the tensor (cross) product or dyad which means one set by two. Particularly, it holds for three arbitrary vectors that

20

1 Tensor Analysis

(u ⊗ v)ir (w )r = ui vr wr which can be expressed in the symbolic notation as

(u ⊗ v)w = u( v • w ) and thus the following expression holds.

u ⊗ v = u( v •

(1.86)

The component of vector is expressed by the direct notation in Eq. (1.38). Here, consider the component of second-order tensor in the direct notation. The second-order tensor T is expressed from Eq. (1.69) as

T = Tij ei ⊗ e j

(1.87)

from which, noting Eq. (1.86) it holds that ei • Te j = ei • Trs e r ⊗ e s e j = Trsδ ir δ sj and thus, the component of T in the direct notation is given as Tij = ei • Te j

(1.88)

As known from Eq. (1.88), the orthogonal projection of the vector Te j to the base vector ei is the component of the tensor T . Especially, Tij (i = j ) and Tij (i ≠ j ) are called the normal component and the shear component, respectively.

1.4 Operations of Second-Order Tensor As described in 1.3.3, tensor operations must be expressed by component notation in general. While the operations Tv , TS , ΞT appear often in the elastoplasticity, let them be meant the following operations.

(Tv )i = Tir vr , (TS)ij = Tir vrj , (ΞT)ij = Ξ ijkl Tkl

(1.89)

Various operations of the second-order tensor are described below.

1.4.1 Trace An operation taking the sum of the components having the same suffixes, i.e. the sum of diagonal components in the matrix notation is called the trace and is expressed as

trT = Trsδ rs = Trr = T11 + T22 + T33

(1.90)

1.4 Operations of Second-Order Tensor

21

tr(TS ) = Tir S ri = T11S11 + T12 S 21 + T13 S 31

+T21T12 + T22T22 + T23T32 +T31T13 + T32T23 + T33T33

(1.91)

The following relations hold for the trace.

t r (T + S) = t r T + t r S, t r (aT) = a t rT, t r (TS) = t r (ST), t r (u ⊗ v )=u • v (1.92)

1.4.2 Various Tensors 1) Zero tensor A tensor, the components of which are all zero, is called the zero tensor and is denoted by the symbol 0 , i.e. (1.93) (0)ij = 0 The zero tensor transforms any tensor to a zero tensor. 2) Identity tensor The tensor which transforms a tensor to itself is called the identity tensor. The second-order identity tensor has the components expressed by the Kronecker’s delta and is denoted by the symbol I , i.e.

I ij = δ ij

(1.94)

The fourth-order identity tensor has the components given by the following equation and is denoted by the symbol I .

I ijkl ≡ 1 (δ ik δ jl + δ ilδ jk ) 2

(1.95)

(TT )ij = T ji

(1.96)

3) Transposed tensor

The tensor TT fulfilling

is called the transposed tensor of T. It holds from {(TS)T }ij = T jr Sri that (TS)T = ST TT

(1.97)

22

1 Tensor Analysis

Further, the following relation holds for the trace.

t r TT = t r T

(1.98)

The magnitude of tensor is defined as the square root of the sum of the squares of each components and thus it is expressed using Eq. (1.96) as T = TijTij = tr (TTT )

(1.99)

The tensor whose magnitude is unity is called the unit tensor. It holds for arbitrary vectors u, v from Eq. (1.96) that

Tu • v = u • TT v,

Tij u j vi = u j (TT ) ji vi

(1.100)

Further, it holds from Eq. (1.100) that

Tu • Sv = u • TT Sv

(1.101)

4) Inverse tensor The tensor T−1 fulfilling the following relation is defined as the inverse tensor of the tensor T . −

1 TT−1 = I , Tir (T ) rj = δ ij

(1.102)

It holds from Eq. (1.15) with Eq. (1.3) for n = 3 that

δ ij det T = 1n δ ijδ ijTi s Δ j s = Ti s Δ j s from which one has T T Δ = I, det T

Tis

Δ js = δ ij det T

(1.103)

Consequently, T −1 is given by −

T1=

ΔT , det T

−

(T 1 )ij =

Δ ji det T

(1.104)

Then, det T ≠ 0 is required in order that T−1 exists, while the tensor fulfilling this condition is called the non-singular tensor. The partial derivative of Eq. (1.23) is rewritten by Eq. (1.104) as

∂ det T = (det T)T−T ∂T

(1.105)

The derivation of Eq. (1.105) starting from the definition of the total differential equation has been often described in some literatures (cf. Leigh, 1964) but it needs cumbersome manipulations. Compared with it, the derivation shown above would be concise and straightforward.

1.4 Operations of Second-Order Tensor

23

The following relation holds for the inverse tensor. −

−

−

−

(TT ) 1 = (T −1)T ( ≡ T T ), (TS) 1 = S 1T

−1

(1.106)

because of

((TT −1 )T =) (T −1 )T TT = I = (TT ) −1 TT −

−

TS(TS) 1 = I → S(TS) 1 = T−1 Now, when we regard the transformation of the vector v to the vector u by the tensor T , i.e.

Tv = u, Tij v j = ui

(1.107)

as the simultaneous equation in which the components of v are the unknown numbers, solution exists for u ≠ 0 if det T ≠ 0 and is given by v = T −T u as T Δ ji v = Δ u, vi = uj detT detT

(1.108)

Here, T must be the non-singular tensor fulfilling det T ≠ 0 in order that the non-trivial solution v ≠ 0 exists for u ≠ 0 . On the other hand, T must be the singular tensor fulfilling det T = 0 in order that the solution v ≠ 0 exists for u = 0. 5) Symmetric and skew-symmetric tensors Tensors TS and T A fulfilling the following relations are defined as the symmetric and the skew (or anti)-symmetric tensor, respectively.

TST = TS , T jiS = TijS

(1.109)

T AT = −T A , T jiA = −TijA

(1.110)

and

An arbitrary tensor T is uniquely decomposed into the symmetric and the skew-symmetric tensors.

T = TS + T A

TS = 1 (T + TT ), T A = 1 (T − TT ) 2 2

(1.111) (1.112)

24

1 Tensor Analysis

while the components of TS and T A are often denoted by T(ij ) and T[ij ] , respectively. Eq. (1.111) is called the Cartesian decomposition, following the decomposition of a complex number to a real and an imaginary parts. It holds that

(T S )T = T S , (T A )T = −T A ((T A )ii = 0, no sum)

(1.113)

and

t r T S = t r T,

t r TA = 0

(1.114)

The diagonal components of the skew-symmetric tensor are zero, and thus its determinant is zero, i.e.

det T A = 0

(1.115)

6) Mean and deviatoric components

When the tensor T is decomposed as follows:

T = Tm + T'

⎫ Tm ≡ TmI, Tm ≡ 1n (tr T) ⎪ ⎬ T' ≡ T − TmI (trT' = 0) ⎭⎪

(1.116)

(1.117)

Tm and T' are called the mean (or spherical) part and the deviatoric part of the tensor T . Noting Eq. (1.113), the skew-symmetric tensor T' A of the deviatoric tensor T' is given by

T' A = T A

(1.118)

Then, the symmetric part of the deviatoric tensor is given by

T' S = T' − T A = T − Tm I − T A

(1.119)

from which one has

T = Tm I + T' S + T A

(1.120)

The decomposition of T into the mean component TmI , the deviatoric symmetric A component T' S and the skew-symmetric component T is called triple decomposition.

1.4 Operations of Second-Order Tensor

25

7) Axial vector The skew-symmetric tensor T A has three independent components in the three-dimensional state. Therefore, vector t A having the following components is called the axial vector.

t Ai = − 1 ε rsiT rsA 2

(1.121)

Inversely from Eq. (1.121) it is obtained that

T ijA = −ε ijr t Ar ,

⎡0 − t A3 t 2A ⎤ ⎢ ⎥ T ijA = ⎢ 0 − t 1A ⎥ ⎢ ant. 0 ⎥ ⎢⎣ ⎥⎦

(1.122)

Furthermore, noting Eq. (1.41) and the relation

T irAvr = −ε irs t sA vr = ε irs t rA vs

(1.123)

the following relation holds.

TAv = t A × v A

tA

(1.124)

The relation of T and is shown in Fig. 1.2 in the case that is the angular velocity vector and v is the position vector of particle. The quantity in Eq. (1.124) tA

tA

|| vv||||sin sinθ θ n

TAv = t A× v = ( || t A |||| v ||sinθ ) n

θ

v

0 Fig. 1.2 Meaning of axial vector in case of rotation

26

1 Tensor Analysis

designates the peripheral velocity vector, while T A is called the spin tensor which induces the peripheral velocity by undergoing the multiplication of the position vector.

1.5 Eigenvalues and Eigenvectors The tensor T is expressed in the component notation having only normal components by choosing the coordinates with special bases. In what follows, consider the special direction for the second-order tensor in the nth dimensional space. The unit vector e fulfilling (1.125)

Tij e j = Tei , Te = Te i.e.

(Tij − Τ δ ij )e j = 0,

(T − Τ Ι )e = 0

(1.126)

for the second-order tensor is called the eigenvector (or principal or characteristic or proper vector) and the scalar T is called the eigenvalue (or principal or characteristic or proper) value. The necessary and sufficient condition that e has a non-zero solution in the simultaneous equation (1.126) is given by

Tij − T δij = 0, det(T − TI) = 0

(1.127)

noting the description in 1.6. Eq. (1.127) is called the characteristic equation of the tensor, which is regarded as the equation of nth degree of T . Unit vectors

eⅠ, eⅡ,… , e N are derived for each of solutions TⅠ, TⅡ,

…, TN

from Eq. (1.126).

Eigen values are real numbers and eigenvectors are mutually orthogonal for eigenvalues that differ from each other in the second-order real symmetric tensor, the components of which are real and which fulfill the symmetry Tij = T ji . The simple proof for this fact is presented below. Regarding T and ei as complex numbers in general and taking the conjugate complex number of both sides in Eq. (1.125), one has the following equation, denoting the conjugate complex number of T and ei by T and ei (note ab = a b for arbitrary complex numbers a, b ).

Tij e j = T ei

(1.128)

Multiplying both sides of Eqs. (1.125) and (1.128) by ei and ei , respectively, and taking the difference in both sides, one has

1.5 Eigenvalues and Eigenvectors

27

Tij e j ei − Tij e j ei = (T − T )ei ei

(1.129)

The left-hand side in Eq. (1.129) becomes zero because of Tij = Tji . Thereby, one has T = T which means that T must be a real number because ei ei ≠ 0 . Furthermore, from the two equations for the two solutions (1.130)

TeⅠ = TⅠⅠ e , TeⅡ = TⅡeⅡ one has

eⅡ • TeⅠ − eⅠ • TeⅡ = (TⅠ − ΤⅡ)eⅠ • eⅡ

(1.131)

The left-hand side of Eq. (1.131) is zero because of the symmetry of the tensor T . Therefore, one has

(TⅠ − Τ Ⅱ)eⅠ • eⅡ = 0

(1.132)

Consequently, it can be concluded that eⅠ and eⅡ are mutually orthogonal for TⅠ ≠ TⅡ . Based on the result described above, denoting the eigenvectors by

ⅠⅡ …, N ) and the corresponding eigenvalues as T , one can write

eJ ( J = , ,

J

TeJ = TJ eJ (no sum)

(1.133)

In addition, noting that the shear component on the coordinate system with the base vector {eJ } is zero, i.e.

TJK (J ≠ K ) = 0

(1.134)

T = TJ e J ⊗ e J

(1.135)

the tensor T is expressed by

If tensor T having eigenvector eJ has the same eigenvalues as tensor T , it holds that

TeJ = TJ eJ (no sum)

(1.136)

where the orthogonal tensor Q between the eigenvectors of these tensors is given by

QIJ = e I • eJ , Q = eJ ⊗ eJ ,

eJ = QeJ , eJ = QT eJ ,

Applying Q to Eq. (1.133), one has

(QT TeJ =) QT TQQ T eJ = TJ QT eJ (no sum)

(1.137)

28

1 Tensor Analysis

from which, considering Eqs. (1.136) and (1.137), it holds that

QT TQ eJ = TJ eJ = TeJ (no sum)

(1.138)

Then, one obtains the relation

T = QT TQ

(1.139)

As presented above, tensors having identical eigenvalues can be related by the orthogonal tensor; they are called the similar tensor mutually. The coordinate transformation rule (1.74) of a certain tensor and the relation (1.139) of two tensors having identical eigenvalues but different eigenvectors, are of mutually opposite forms. If the function f of tensor A, B, • • • is observed to be identical independent of observers, i.e. if it fulfills the relation f ( A, B, • • • ) = f ( Q A , Q B , • • • )

(1.140)

using the symbol in Eq. (1.71), f is called the isotropic scalar-valued tensor function, which is none other than the invariant. In particular, The isotropic

scalar-valued tensor function f (T) of single second-order tensor T fulfills

f (T) = f (QTQT )

(1.141)

f (T) is expressed by three principal values in the three-dimensional case, involving them in symmetric form so as to be identical even if they are exchanged to each other. Then, there exist three independent invariants for a single tensor. Their explicit forms are presented below. The expansion of the characteristic equation (1.127) of T for n = 3 leads to

T11 − Τ

T12

T21 T22 − Τ T31

T32

T13 T23 T33 − Τ

= (T11 − Τ )(T22 − Τ )(T33 − Τ ) + T12T23T31 + T21T32T13

− (T11 − Τ )T23T32 − (T22 − Τ )T31T13 − (T33 − Τ )T12T21 = −Τ 3 + (T11 + T22 + T33 )Τ 2 − (T11T22 + T22 T33 + T33T11 )Τ + T11T22T33 + 2T12T23T31 + (T12T21 + T23T32 + T31T13 )T − T11T23T32 − T22T31T13 − T33T12T21 = −Τ 3 + (T11 + T22 + T33 )Τ 2 − (T11T22 + T22 T33 + T33T11 − T12T21 − T23 T32 − T31T13 )Τ + T11T22 T33 − T11T23T32 − T22 T31T13 − T33T12T21 + 2T12T23T31

1.5 Eigenvalues and Eigenvectors

29

= − Τ 3 + ( T11 + T22 + T33 )Τ 2

−

1 [{Τ 112 + Τ 222 + Τ 332 + 2(T11T22 + T22 T33 + T33T11 )} 2

− {Τ112 + Τ 222 + Τ332 + 2(T12T21 + T23T32 + T31T13 )}] Τ + T11T22 T33 + T12T23T31 − T11T23T32 − T22 T31T13 − T33T12T21 + 2T12T23T31

= −Τ 3 + (T11 + T22 + T33 )Τ 2 −

1 [( T11 + T22 + T33 ) 2 − {Τ 112 + Τ 222 + Τ 332 + 2 ( T12T21 + T23T32 + T31T13 )}] Τ 2

+ T11T22T33 − T11T232 − T22T312 − T33T122 + 2T12T23T31 = 0 from which the characteristic equation is given as

Ⅱ Ⅲ= 0

(1.142)

+ T22 + T33 = Tii = trT

(1.143)

T 3 −I T 2 + Τ − where

Ⅰ≡ T

11

Ⅱ ≡ TT

11

T12

21

T22

+

T22 T23 T32 T33

+

T11 T13 T31 T33

1 1 = Dii = Δii = (TrrTss − TrsTsr )= ( tr 2 T − trT 2 ) 2 2 (1.144)

T11 T12 T13

Ⅲ≡ T

1 1 1 3 T22 T23 = detT = ε rstTr1Ts 2Tt 3 = 6 tr 3T − 2 tr T trT 2 + trT 3 T31 T32 T33 21

(1.145) The direct notation of III is derived by taking the trace of the expression of the Cayley-Hamilton theorem described in the next section and substituting the direct notations of I and II into it. On the other hand, the characteristic equation (1.142) is expressed using the principal values as follows:

(T − TⅠ)(T − TⅡ)(T − TⅢ ) = 0

(1.146)

Comparing Eqs. (1.151) and (1.155), coefficients I, II, and III are described as

Ⅰ= TⅠ+TⅡ+TⅢ ⎫ Ⅱ = TⅠTⅡ + TⅠTⅡ + ΤⅢTⅠ⎪⎬ ⎪ Ⅲ=TⅠTⅡTⅢ ⎭

(1.147)

30

1 Tensor Analysis

Eq.

(1.147)

can

also

be

derived

by

substituting

T11 = T , T22 = T ,

T33 = TⅢ , T12 = T23 = T31 = 0 in the determinants in Eqs. (1.143)-(1.145). Since I, II, and III are the symmetric functions of principal values, they are the invariants and are called the principal invariants. Next, consider the deviatoric tensor T' . The characteristic equation of T' is given by replacing T to T' in Eq. (1.142) as follows:

T ' 3 − 'ȉ ' − ' = 0

(1.148)

Ⅰ' ≡ trT' = 0

(1.149)

noting

where

Ⅱ' ≡ Dii' = Δii' = 12 trT'

2

1 = 2 Trs' Tsr'

1 = 2 (T11' 2 + T22' 2 + T33' 2 ) + T12' 2 + T23' 2 + T31' 2 2 2 2 1 = {(T11 − T22 ) 2 + (T22 − T33 ) 2 + (T33 − T11 ) 2 } + T12' + T23' + T31' 6

1 1 = 2 (TⅠ' 2 + TⅡ' 2 + TⅢ' 2 ) = {(TⅠ − TⅡ) 2 + (TⅡ − TⅢ ) 2 + (TⅢ − TⅠ) 2} 6

(1.150)

and

T11' T12' T13'

Ⅲ' ≡ T'

21

1 1 3 T22' T23' = det T' = 3 trT' = 3 Trs' Tsr' Trt'

T31' T32' T33'

= T11' T22' T33' − T11' T23' − T22' T31' 2 − T33' T12' 2 + 2T12' T23' T31' 2

1 = TⅠ'TⅡ'TⅢ' = 3 (TⅠ' 3 + TⅡ' 3 + TⅢ' 3)

(1.151)

The direct notation in (1.151) is derived by taking the deviator and the trace of the expression of Cayley-Hamilton’s theorem described in the next section and substituting the direct notations of ' (=0) and ' in Eqs. (1.149) and (1.150) into that.

Ⅰ

Ⅱ

1.6 Calculations of Eigenvalues and Eigenvectors

31

1.6 Calculations of Eigenvalues and Eigenvectors The symmetric tensor T can be represented in the eigendirections as N

T = ∑ TJ eJ ⊗ eJ

(1.152)

J =1

which is called the spectral representation. To express the tensor in the eigendirections, one must calculate the eigenvalues and the eigenvectors of the tensor. The solutions for them (cf. Hoger and Carlson, 1984 and Carlson and Hoger, 1986) are shown in this section.

1.6.1 Eigenvalues In order to obtain eigenvalues, one must solve the characteristic equation which is the cubic equation having the coefficients as the functions of invariants. Now, infer the form

Ⅱ

4 ' (1.153) cosψ 3 for the eigenvalues of deviatoric part of tensor T . The substituting Eq. (1.153) into Eq. (1.148), we have T' =

') ( 4Ⅱ 3

3/ 2

Ⅱ Ⅱ

cos3 ψ − ' ( 4 ' ) 3

1/2

Ⅲ

cosψ − ' = 0

(1.154)

which reduces to

Ⅲ

Ⅱ

4 3 / 2 cos 3ψ − ' =0 ' 3 3

(1.155)

using the trigonometric formula

cos 3 ψ = 1 (cos 3ψ + 3 cosψ ) 4 It is obtained from (1.155) that

(1.156)

Ⅲ

(1.157) cos 3ψ = 3 3 3/ 2' 2 ' Noting that the cosine is the periodic function with the period 2π , the angle ψ is expressed by the following equation in general.

Ⅱ

Ⅲ

ψ J = 1 {cos −1 (3 3 3/ 2' ) − 2π J } 3

Ⅱ

2 '

(1.158)

32

1 Tensor Analysis

Ⅰ

Substituting Eq. (1.158) into Eq. (1.153) and adding the isotropic component / 3 , the eigenvalues of T are given as follows:

TJ = 1 3

(Ⅰ+ 4Ⅱ3 ' cos[13{cos (32Ⅱ3'Ⅲ' ) − 2π J }]) −1

3/ 2

(1.159)

1.6.2 Eigenvectors Eq. (1.152) can be expressed as follows: N

T = ∑ TJ EJ

(1.160)

J =1

while the tensor EJ is called the eigenprojection of T , which is defined by EJ ≡ eJ ⊗ eJ (no sum)

(1.161)

fulfilling N

E J (= eⅠ⊗ eⅠ+ eⅡ ⊗ eⅡ + …+ e m ⊗ em ) = I ∑ J

(1.162)

=1

⎧E J for J =K EJ E K = ⎨ ⎩ 0 for J ≠ K tr(E J E K ) = δ IJ

⎫ ⎪ ⎬ ⎪ ⎭

(1.163)

It holds that N N ⎫ TEJ = ( ∑ TK EK )e J ⊗ e J = ( ∑ TK e K ⊗ e K) e J ⊗ e J = TJ e J ⊗ e J ⎪ ⎪ K =1 K =1 ⎬ (no sum for J ) N N E J T = e J ⊗ e J ( ∑ TK E K ) = e J ⊗ e J ( ∑ TK e K ⊗ e K) = e J ⊗ eJ TJ ⎪ ⎪⎭ K =1 K =1

and thus one has

TE J = EJ T = TJ EJ (no sum for J ) On the other hand, it holds from Eq. (1.162) that N

T − TK I = ∑ TJ EJ − ΤK J =1

N

EJ ∑ J= 1

(1.164)

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

33

and thus it is obtained that N

T − TK I = ∑ (TJ − ΤK ) E J

(1.165)

J =1

from which one has N

∏ K ≠θ

N

N

∑ (TJ − ΤK ) E J K ≠θ J =

(T − TK I ) = ∏

K =1

(1.166)

1

K =1

The right-hand side in Eq. (1.166) is rewritten as N

N

N

∑ (TJ − ΤK ) EJ = {∏ (Tθ − ΤK )}Eθ

∏ K ≠θ J = K =1

(1.167)

K ≠θ K =1

1

noting Eq. (1.163). Then, considering E θ = I for the case of N = 1 , i.e. a single root, the following Sylvester’s formula is obtained.

⎧ N T − TK I for N >1 ⎪∏ E θ = ⎨ K ≠θ Tθ − TK K =1 ⎪ for N =1 I ⎩

(1.168)

For instance, E2 (θ =2) in the popular case of N = 3 is obtained by the above-mentioned method as follows:

∏ (T − TK I ) = (T − T1I )(T − T3I )

K =1, 3

∏

3

(TJ − ΤK ) E J ∑ J=

K =1, 3

1

={( T1 − Τ1) E1 + ( T2 − Τ1) E2 + ( T3 − Τ1) E 3} {( T1 − Τ3 ) E1 + ( T2 − Τ3 ) E2 + ( T3 − Τ3 ) E 3} =( T2 − Τ1)( T2 − Τ3 )E2 = { ∏ (T2 − ΤK )}E 2 = ( T2 − Τ1)( T2 − Τ3 ) E 2 K =1, 3

∏ (T − TK I )

E2 =

K =1, 3

∏ (T2

K =1, 3

− TK)

=

(T − T1I )(T − T3I ) ( T2 − Τ1)( T2 − Τ3 )

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor The characteristic equation of skew-symmetric tensor is given by substituting TA into Eq. (1.142) as follows:

34

1 Tensor Analysis

1 2 2 T 3 − trT AT 2 + 2 ( tr T A − trT A )Τ − detT A = 0

(1.169)

Noting trT A = detT A = 0 in Eqs. (1.114) and (1.115), Eq. (1.169) leads to 2 T (2T 2 − trT A ) 2 = 0

(1.170)

from which we have 2

T = ±i | trT A | / 2 = ± i|| t A || and 0

(1.171)

noting

trT A 2 = −2( t 1A 2 + t 2A 2 + t 3A 2 ) < 0 obtained from (1.122). It is known from Eq. (1.171) that the real eigenvalue of skew-symmetric tensor is zero. Here, if select one of the principal direction e Ⅲ to the one with the zero principal value, it holds that T A * = [Q][T A ][Q]T

⎡ cos θ sinθ 0 ⎤ ⎡ 0 ω 0 ⎤ ⎡ cosθ _sinθ 0 ⎤ = ⎢⎢ _sinθ cosθ 0 ⎥⎥ ⎢⎢ −ω 0 0 ⎥⎥ ⎢⎢sinθ cos θ 0 ⎥⎥ ⎢⎣ 0 0 1 ⎥⎦ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 1 ⎥⎦

⎡ −ωsinθ cos θ + ωsinθ cos θ ω sin 2 θ + ω cos 2 θ 0⎤ ⎢ _ ⎥ 2 2 =⎢ ω cos θ − ωsin θ ωsinθ cos θ − ω sinθ cosθ 0 ⎥ ⎢ 0 0 1 ⎥ ⎣ ⎦ ⎡ 0 ω 0⎤ = ⎢⎢ −ω 0 0 ⎥⎥ = T A ⎢⎣ 0 0 1⎥⎦

(1.172)

meaning that the components do not change in the coordinate transformation. It is caused from the fact that the independent component of skew-symmetric tensor is only one when the one of base in the coordinate system is chosen to the principal direction of skew-symmetric tensor.

1.8 Cayley-Hamilton’s Theorem

35

1.8 Cayley-Hamilton’s Theorem Denoting the eigenvector of the tensor T by unit vector e , it holds from Eq. (1.125) and T −1e = T −1Te / T = T −1Te / T = eT −1 that

Tr e = T r e

(1.173)

for r = ±1 . Now, assuming that Eq. (1.173) holds for r = k , one has

Tk ±1e = Tk T±1e = TkT ±1e = T ±1Tk e = T ±1T k e = T k ±1e Consequently, Eq. (1.173) holds also for r = k ± 1 . Then, it is verified that Eq. (1.173) holds for all integers r by mathematical induction. Eq. (1.173) means that the principal value of T r is T r and the principal directions of the tensors T and T r are identical mutually, while the tensors having an identical set of principal directions are called to be coaxial or said to fulfill the coaxiality. Then, the linear associative function f (T) of T is coaxial with T and the principal values are given by f (T ) . For n = 3 , the multiplication of the eigenvector e to the characteristic equation (1.142) leads to

Ⅰ Ⅱ ⅢI )e = 0

( T3 − T 2 + T −

noting Eq. (1.173). Because of e ≠ 0 , the following Cayley-Hamilton theorem holds.

Ⅰ Ⅱ Ⅲ

T3 − T2 + T − I = 0

(1.174)

It follows from the Cayley-Hamilton theorem that

Ⅰ Ⅱ Ⅲ Ⅰ Ⅱ ⅢT =Ⅰ(ⅠT −ⅡT + ⅢI) −ⅡT + ⅢT = (Ⅰ −Ⅱ)T − (ⅠⅡ − Ⅲ)T +ⅠⅢI (1.175) (1.176) Ⅲ T = T −ⅠT + ⅡI

T4 = ( T2 − T + I)T = T3 − T2 + 2

2

2

2

−1

2

It is concluded that the power of the tensor T is expressed by the linear associative of T2, T, I with coefficients consisting of the principal values.

1.9 Positive Definite Tensor When the second-order tensor P is symmetric and fulfills

Pv • v > 0 for an arbitrary vector v(≠ 0) , P is called the positive definite tensor.

(1.177)

36

1 Tensor Analysis

Denoting the principal value and direction of P as PJ and e J , respectively, it holds that

Pe J • e J = PJ e J • e J = PJ || e J || 2 > 0 (no sum)

(1.178)

noting Eq. (1.177). Then, it is known that the principal value of positive definite tensor is positive. Taking this fact into account for Eq. (1.147)3, it holds that det P =

Ⅲ> 0

(1.179)

The positive definite tensor U having the same principal directions and principal 1/ 2 values PJ is defined as the square root of P , i.e. U 2 = P or U = P1/ 2 .

1.10 Polar Decomposition Assuming that the second-order tensor T is not singular ( det T ≠ 0 ), it holds that Tv ≠ 0 for an arbitrary vector v (≠ 0) as described in 4) of 1.4.2 and thus using

Eq. (1.101), one obtains

TT Tv • v = Tv • Tv > 0

(1.180)

where TT T is the symmetric tensor and thus it is the positive-definite tensor as described in 1.9. Denoting the square root of TT T by U , one can write U = (TT T)1/2 (U 2 = TT T), UT = U

(1.181)

Then, U is the positive definite tensor. Furthermore, defining the tensor R as R = TU −1

(1.182)

noting Eq. (1.106), one has

RRT = (TU −1 )(TU −1 )T = TU −1U −1TT = T(U 2 ) −1 TT = TT−1T−T TT = I (1.183) Therefore, R is the orthogonal tensor. Furthermore, similarly to Eqs. (1.181) and (1.182), consider V = (TTT )1/ 2 (V2 =TT T)

(1.184)

from which we have V 2 = TTT = ( RU ) ( RU )T = RUUR T = RUR T RUR T = ( RUR T ) 2

(1.185)

1.11 Isotropic Tensor-Valued Tensor Function

37

noting Eq. (1.182), and thus it holds that V = RURT , U=RT VR

(1.186)

T = RU = VR

(1.187)

Then, one can write

Consequently, an arbitrary non-singular tensor T can be decomposed into two forms in terms of the positive definite tensors U or V and the orthogonal tensor R . Here, based on Eqs. (1.181), (1.182), (1.184), and (1.185), R is expressed by the original tensor T as follows: R = (TTT )−1/ 2 T = T(TT T)−1/ 2

(1.188)

Based on (1.186), U, V are the mutually similar tensors, as described in 1.5.1. For that reason, they have same positive principal values, denoted as U , U , U , and

Ⅰ Ⅱ Ⅲ

ⅠⅡⅢ) are mutually related by

their unit principal vectors uJ , vJ ( J = , ,

vJ = Ru J , R =vJ ⊗ uJ

(1.189)

according to Eqs. (1.137), (1.139) and (1.186). Equation (1.187) is called the polar decomposition in similarity to the polar form Z = Z eiθ (θ : phase angle) which expresses the complex number by the decomposition into the magnitude and the direction in the polar coordinate system.

Actually, RU and VR are respectively called the right and the left polar decompositions.

1.11 Isotropic Tensor-Valued Tensor Function If the function f of tensors T, S, • • • fulfills the following equation, it is called the isotropic function. Q f (T, S, • • •) = f ( Q T , Q S , • • • )

(1.190)

where the symbol in Eq. (1.72) is used. If f is a scalar, it is to be the invariant defined in Eq. (1.140) and if it is a tensor, it is called the isotropic tensor-valued tensor function.

Now, consider the isotropic second-order tensor function B of a single second-order tensor A , i.e.

38

1 Tensor Analysis

B = f ( A)

(1.191)

where f fulfills f (QAQT ) = Qf ( A)QT

(1.192)

First introducing the coordinate system with the bases e I , eⅡ, eⅢ , which are the normalized eigenvector of the tensor A and further adopting the another coordinate system rotated 180°around the base eⅢ , the orthogonal tensor between the bases of these coordinate systems is given by ⎡ −1 0 0 ⎤ Q 0 = ⎢⎢ 0 − 1 0 ⎥⎥ ⎢⎣ 0 0 1⎥⎦

(1.193)

where Q0 fulfills Q 0 = QT0 resulting in the symmetric tensor and it holds that

⎡ − 1 0 0 ⎤ ⎧0 ⎫ ⎧0 ⎫ ⎢ 0 − 1 0 ⎥ ⎪0 ⎪ = ⎪0 ⎪ , i.e. Q0eⅢ = eⅢ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎢⎣ 0 0 1⎥⎦ ⎪⎩1 ⎪⎭ ⎪⎩1 ⎪⎭

(1.194)

Then, it is known that eⅢ is one eigenvector not only of A but also of Q 0 . Furthermore, denoting the principal values of A by αⅠ, αⅡ, αⅢ , it holds that

⎡ −1 0 0 ⎤ ⎡αⅠ 0 0 ⎤ ⎡ −1 0 0 ⎤ ⎡αⅠ 0 0 ⎤ ⎢ 0 − 1 0 ⎥ ⎢ 0 α 0 ⎥ ⎢ 0 − 1 0 ⎥ = ⎢ 0 α 0 ⎥ , i.e. Q AQ T = A Ⅱ ⎥⎢ Ⅱ ⎥ 0 0 ⎢ ⎥⎢ ⎥ ⎢ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 αⅢ⎥⎦ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 αⅢ⎥⎦ (1.195) and thus it holds that f (Q 0 AQ 0T ) = f ( A ) = B

(1.196)

On the other hand, from Eq. (1.192) one has f (Q 0 AQ T0 ) = Q 0 BQ T0

(1.197)

Then, the commutative law Q 0 B = BQ 0

(1.198)

1.11 Isotropic Tensor-Valued Tensor Function

39

holds from Eqs. (1.196) and (1.197), and further, noting Eq. (1.194), the following relation is obtained.

Q 0 BeⅢ = BQ 0eⅢ = BeⅢ

(1.199)

which means that B eⅢ is the eigenvector of Q 0 and thus it has the same direction as eⅢ . Then, denoting the principal value B for the eigenvector eⅢ by βⅢ , one can writes

BeⅢ = βⅢ eⅢ

(1.200)

Performing the similar manipulations also for eⅠ and eⅡ , it can be concluded that the tensor B has the same eigenvectors as the tensor A , leading to the coaxility. Therefore, the principal values βⅠ, β Ⅱ, β Ⅲ of the tensor B can be represented in unique relation to the principal values αⅠ, αⅡ, αⅢ of the tensor A . Now, noting the results obtained above, i.e. the coaxiality and the representation of βⅠ, β Ⅱ, β Ⅲ by αⅠ, αⅡ, αⅢ in view of the coaxiality, consider the following equation by way of trial.

βⅠ(αⅠ, αⅡ, αⅢ ) = φ0 + φ1αⅠ + φ2αⅠ2 ⎫

⎪⎪ βⅡ(αⅠ, αⅡ, αⅢ ) = φ0 + φ1αⅡ + φ2αⅡ2 ⎬

(1.201)

⎪

β Ⅲ(αⅠ, αⅡ, αⅢ ) = φ0 + φ1α Ⅲ + φ2α Ⅲ2 ⎪⎭

where φ0 , φ1 , φ2 are symmetric functions of αⅠ, αⅡ, αⅢ . While Eq. (1.201) is regarded as the representation of the relation of the tensors A and B in their common principal coordinate system, it is expressed in the direct notation of tensor as

B = φ0 I + φ1A + φ2 A 2

(1.202)

in which φ0 , φ1 , φ2 are functions of invariants of the principal tensor A . When we regard Eq. (1.201) to be the simultaneous equation for the unknown values φ0 , φ1 , φ2 , we know that the Vandermonde’s determinant is not zero for mutually different values of αⅠ, αⅡ, αⅢ as follows:

1 αⅠ αⅠ2

1 αⅡ αⅡ2 = (αⅠ− αⅡ)(αⅡ − αⅢ )(αⅢ − αⅠ) ≠ 0 1 αⅢ α Ⅲ2

(1.203)

40

1 Tensor Analysis

Therefore, φ0 , φ1 , φ2 are scalar functions which are uniquely determined by αⅠ, αⅡ, αⅢ and then βⅠ, β Ⅱ, β Ⅲ are uniquely determined by αⅠ, αⅡ, αⅢ , while φ0 , φ1 , φ2 are invariants of A since they are scalar functions of αⅠ, αⅡ, αⅢ , i.e. A . Thus, we can conclude that Eq. (1.202) is the correct explicit form of Eq. (1.191). Although Eq. (1.202) would not be the unique explicit form of Eq. (1.191), one can say that the isotropic tensor-valued tensor function of a single tensor reduces to this simplest form. Eventually, Eq. (1.202) is the most concise from of the isotropic second-order tensor-valued function of a single second-order tensor. This fact can also be verified using Cayley-Hamilton’s theorem for the special case that f is the linear associative form of the power of A . However, for the case in which f is the general function of A , one must depend on the proof given in this section. In the particular case in which f is the linear function of the tensor A , Eq. (1.202)reduces to

B = a (trA)I + bA where

(1.204)

a and b are the material constants. Equation (1.204) is rewritten as B = CA

(1.205)

C ≡ aI ⊗ I + b I (Cijkl ≡ aδ ijδ kl + 1 b(δ ikδ il + δ ilδ jk ) 2

(1.206)

where

While the second-order isotropic tensor-valued tensor function of single tensor is considered above, the representation theorem of the second-order isotropic tensor-valued tensor function f of two tensors A and B is sown below (cf. e.g., Spencer, 1971).

f(A , B) = ϕ 0 I + ϕ1 A + ϕ 2 B 2 + ϕ 3 A 2 + ϕ 4 B 2 + ϕ 5 ( AB + BA )

+ ϕ6 ( A 2B + BA 2 ) + ϕ7 ( AB 2 + B 2 A ) + ϕ8 ( A 2 B 2 + B 2 A 2 )

(1.207)

where ϕ0 , ϕ1 , • ••, ϕ8 are scalar functions of invariants

⎪⎫ ⎬ tr( AB), tr( AB ), tr( A B), tr( A B ) ⎪⎭ trA, trA 2 , trA3 , trB, trB 2 , trB3 2

2

2

(1.208)

2

1.12 Representation of Tensor in Principal Space Let the second-order tensor T be expressed by the vector from in terms of the

TI , TⅡ, TⅢ T = TI eI + TⅡeⅡ + TⅢeⅢ

eigenvectors eI , eⅡ, eⅢ and the principal values

as follows: (1.209)

1.12 Representation of Tensor in Principal Space

41

Equation (1.209) is called the representation of tensor in principal space (Fig. 1.3) by which the second-order tensor can be visualized in the three dimensional space. Equation (1.209) is rewritten by decomposing T into the mean and the deviatoric components as follows:

T = (Tm + TI' )eI + (Tm + TⅡ' )eⅡ + (Tm + TⅢ' )eⅢ = Tm + T'

(1.210)

Tm ≡ Tm I m = 3Tmem (Tm ≡ (TI + TⅡ + TⅢ )/3)

(1.211)

I m ≡ e I + eⅡ + eⅢ , e m ≡ 1 I m (||e m || = 1) 3

(1.212)

where

T' ≡ TI' eI + TⅡ' eⅡ + TⅢ' eⅢ = || T' || t'

TI' ≡ TI − Tm , TⅡ' ≡ TⅡ − Tm , TⅢ' ≡ TⅢ − Tm

(1.213) (1.214)

⎫ ⎪ 1 ( − ) 2 + ( − ) 2 + ( − )2 ⎬ TI TⅡ TⅡ TⅢ TⅢ TI ⎪ || T' || = TI' 2 + TⅡ' 2 + TⅢ' 2 = 3 ⎭

t' ≡ T' / || T' || (||t' || = 1)

(1.215)

T

Space diagonal

T

T' TT' t' T'

TTmm

e

0

e

Deviatoric plane

T'

emm TT e T

Fig. 1.3 Representation of second-order tensor in principal space

42

1 Tensor Analysis

T

0 _

T'

T

_

T'

θ T'

T

_

( −) T '

Fig. 1.4 Coordinate system in deviatoric (Ǹ) plane

Whereas the deviatoric tensor T' lies on the π - plane (Fig. 1.4), the orthogonal projection of T' to the three oblique axes, which are the orthogonal projections of the orthogonal axes TI , TⅡ, TⅢ to the π - plane, are given as

⎫ ⎪ ⎪ ⎪ TⅡ' = || T' ||cos θ − 2 π , ⎬ 3 ⎪ ⎪ 2 T TⅢ = || ||cos θ π + ' ' 3 ⎭⎪ TI' = || T' ||cos θ ,

(

)

(

)

(1.216)

On the other hand, the deviatoric components TI' , TⅡ' , TⅢ' are the components on the orthogonal coordinates (TI , TⅡ, TⅢ ) of the diviatoric tensor T' (see Fig. 1.3). Denoting the angle contained between the coordinate axis, e.g. eI and its projected line onto the π - plane by the symbol α (see Fig. 1.5), it holds that

1.12 Representation of Tensor in Principal Space

43

0ABC: Rectangular triangular pyramid

A

T' C

em

TII'

T

_

T ' = || T' || cos θ T' ≡ T' e + T' e + T' e

e II θ α

0

_

_

TT ' B cos α =

Fig. 1.5 Relation of

T ' = T ' cos α

-plane ππ-plane

2/3

T1' and TI' = e m • eI = 1 , e m • eI = cos(π − α ) 2 3

(1.217)

resulting in

cosα = 2/3

(1.218)

Substituting Eq. (1.218) into Eq. (1.216) it is obtained that

⎫ ⎪ ⎪ ⎪ TⅡ' = 2T ' = 2 || T' ||cos θ − 2 π , ⎬⎪ 3Ⅱ 3 3 ⎪ ⎪ 2 2 TⅢ' = TⅢ = || T' ||cos θ + 2 π ⎪ ' 3 3 3 ⎪⎭ TI' =

2T = 3 ' I

2 || T ||cos θ , 3 '

(

)

(

)

(1.219)

The product of the three components is given by TⅡ' TⅢ ' = 2 2 cos θ cos θ − 2 π cos θ + 2 π = 1 cos 3θ 3 3 3 6 || T' || || T' || || T' || 3 3 TI'

(

) (

)

(1.220) Considering Eq. (1.151) to Eq. (1.220), it follows that

cos 3θ = 6 tr t' 3

(1.221)

44

1 Tensor Analysis

It holds from Eq. (1.216) that

μ≡

2TⅡ − TI − TⅢ 2TⅡ' − TI' − TⅢ' = = 3 tan (θ − 1 π ) 6 TI' − TⅢ' TI − TⅢ

(1.222)

which is called the Lode’s variable. Assuming the principal value T1' =

2 || T || cos θ 3 '

(1.223)

Ⅰ

and substituting it into in the characteristic equation of deviatoric tensor (1.148) while, taking account of ' = 0 , one has the following equation

(

2 || T || cos θ 3 '

)

3

− 1 || T' || 2 2

(

)

2 || T || cos θ − 1 t r T = 0 '3 3 3 '

resulting in

4cos 3θ − 3cos θ − 6 t r t' 3 = 0 from which one can derive Eq. (1.221) and further reach Eq. (1.219) by tracing the above-described process inversely.

1.13 Two-Dimensional State Consider the two-dimensional state in which the components related to the e3 direction in the coordinate system ( x1 , x2 , x 3 ) with the bases

(e1 , e 2 , e3 ) are zero,

i.e. T33 = T 31 = T23 = 0 . Furthermore, introduce the coordinate system

( x1* , x*2 , x3* ) with the bases (e*1 , e*2 , e*3 ( = e3 )) which is rotated by the angle α in the counterclockwise direction around the axis

x3 as shown in Fig. 1.6. The

orthogonal tensor between these bases is given from Eq. (1.47) as follows:

cos α sinα 　 0⎤ ⎡　 ⎢ [Q] = ⎢ −sinα cosα 　 0 ⎥⎥ ⎢⎣ 0 0 1 ⎥⎦

(1.224)

1.13 Two-Dimensional State

x22

45

x2

Tt

Material Material

x1*

(T11*, T12* )

x**22 x11**

R RT

(T22 , T12 )

α

22α

x*2

x11

T

22α pp

α

0 P

T11 + T22 2

T

(T11 , T12 )

Tn

x1

(T22* , T12* ) (a) Physical plane

(b) (Tn , Tt ) plane

Fig. 1.6 Mohr’s circle

Substituting Eq. (1.224) into Eq. (1.62), one has

⎫ ⎪ ⎪ T22* = T11 sin 2α + T22 cos 2α − 2T12 sin α cos α ⎬ ⎪ T12* = (T22 − T11 ) cos α sin α + T12 (cos 2α − sin 2α ) ⎪ ⎭ T11* = T11 cos 2α + T22 sin 2α + 2T12 cos α sin α

(1.225)

which is rewritten as

T11* = Tm + T cos 2α + T12 sin 2α ⎫ ⎪ ⎪ * T22 = Tm − T cos 2α − T12 sin 2α ⎬ ⎪ T12* = −T sin 2α + T12 cos 2α ⎪⎭

(1.226)

where

Tm ≡

T11 + T22 T −T , T ≡ 11 22 2 2

(1.227)

Furthermore, it holds from Eq. (1.226) that

T11* + T22* = T11 + T22

(1.228)

∂T11* ∂T22* = 2T12*, = −2T12* ∂α ∂α

(1.229)

46

1 Tensor Analysis

While the axis x3 (= x3* ) is one of the principal directions, the other principal directions exist on the plane (x1, x2 ) . Denoting the principal direction from the

x1 - axis by α , it is obtained by putting ∂T11* / ∂α = T12* = 0 or ∂T22* / ∂α = 0 in Eq. (1.229) with Eq. (1.226) that

T12 T

tan 2α p =

(1.230)

from which one obtains

T = ± RT cos 2α p , RT = T + Τ 2

2 12

T12 = ± RT sin 2α p ⎫ ⎪ ⎬ ⎪⎭

(1.231)

Substituting Eq. (1.231) into the upper two of Eq. (1.226) with specifying α as α p , the maximum and the minimum principal values TⅠ and TⅡ are described by

TⅠ ⎫ ⎬ = Tm ± R TⅡ ⎭

(1.232)

Equation. (1.232) can also be derived directly from the quadratic equation

T 2 − (T11 + T22 )T + T11T22 − T122 = 0 which

is

obtained

by inserting (T33 = T31 = T23 = 0) in Eq. (1.142).

= T11 + T22 ,

Ⅲ= 0

= T11T22 − T122 ,

Furthermore, denoting α for the extremum of T12* as α s , it holds by taking

∂T12* / ∂α = 0 in Eq. (1.226) that (1.233)

tan 2α s = − T 2T12 Equations

(1.230)

and

(1.233)

yield

the

relation

αs = α p ± π/4

( tan 2α p tan 2α s = −1 ）and thus there exist the two directions for the extremum of

T12* and they divide the two principal directions into two equal angles, i.e. π /4 . The extremum of T12* denoted by TM is given from Eq. (1.231)2 as follows:

TM = ± RT

(1.234)

1.14 Partial Differential Calculi

47

which is also expressed by Eq. (1.232) as

T −T TM = ± Ⅰ Ⅱ 2

(1.235)

Designating the normal stress Tij∗ (i, j = 1, 2; i = j ) by Tn and the shear stress

Tij∗ (i ≠ j ) by Tt , the following equation is derived from Eqs. (1.226) and (1.231)3.

(Tn − Tm ) 2 + Tt = RT2

(1.236)

Consequently, the stress on an arbitrary plane is expressed by the point on the circle with the radius RT centering at (Tm , 0) in the two-dimensional plane (Tn , Tt ) as shown in Fig. 1.5. This circle is called the Mohr’s circle. Substituting Eq. (1.231) into Eq. (1.226), we have the expressions

T11* = Tm + RT cos(2α − 2α p ), ⎫ ⎪ ⎪ * T22 = Tm − RT cos(2α − 2α p ), ⎬ ⎪ T12* = − RT cos(2α − 2α p ) ⎪ ⎭

(1.237)

Therefore, T11* and T12* are shown by the values in the ordinate and abscissa axes, respectively, of the point rotated 2α counterclockwise from point T11 , T12 on the Mohr’s circle as shown in Fig. 1.5, provided that the definition for the sign of shear stress is altered to be positive when it applies to the body surface in the clockwise direction, in the Mohr’s circle. As shown in Fig. 1.5, the intersecting angle of the two straight lines drawn parallelly to the physical plane x1 and x1* stemming from the points (T11 , T12 ) and

(T11*, T12* ) , respectively, on the Mohr’s circle is α which is the angle of circumference of Mohr’s circle and thus the intersecting point lies on the circle. This point is called the pole. Generally speaking, the normal stress Tn and the shear stress

Tt applying to a certain physical plane are given by the intersecting point of Mohr’s circle and the straight line drawn parallel to that physical plane from the pole.

1.14 Partial Differential Calculi Partial derivatives of symmetric tensors appearing often in elastoplasticity are shown below.

48

1 Tensor Analysis

∂Tij 1 = (δ ik δ jl + δ il δ jk ) , ∂ T = I ∂T ∂Tkl 2

(1.238)

1 ∂ ( 1 tr T) ∂Tm ∂ ( 3 Trsδ rs ) 1 3 1 = = 3 δ irδ jsδ rs = δ ij , = 1 I (1.239) 3 3 ∂Tij ∂Tij ∂T

∂Tij' ∂ (Tij − Tmδ ij ) 1 1 = = (δ ik δ jl + δ ilδ jk ) − 1 δ ijδ kl , ∂ T' = I − I ⊗ I 3 2 3 T ∂ ∂Tkl ∂Tkl (1.240)

∂ Trs' Trs' ∂ (Trs' Trs' ) 1 = 1 (Trs' Trs' )−1/ 2 = (Trs' Trs' ) −1/ 2 2δ irδ jsTrs' = t'ij , 2 2 ∂Tij' ∂Tij' ∂ ||T'|| = T' ≡ t' ||T'|| ∂ T'

Tij'

∂ ∂tij' = ∂Tkl'

T pq ' Tpq ' ∂Tkl' 1

=

T'pqT'pq

δ ik δ jl T pq ' Tpq ' − Tij' =

(1.241)

Tkl' T'pqT'pq

T pq ' T'pq

{12 (δikδ jl + δilδ jk ) − tij' tkl' }, ∂ t' = 1 ( I − t' ⊗ t' ) ∂ T' ||T'||

∂tij' ∂tij' ∂Trs' = = ∂Tkl ∂Trs' ∂Tkl'

=

1 Tpq ' T pq '

1

Tpq ' Tpq '

(1.242)

{12 (δ irδ js + δ isδ jr ) − t'ijtrs' }(δ rk δ sl − 13 δ rsδ kl )

{12 (δik δ jl + δilδ jk ) − 13 δijδ kl − tij' tkl' }, ∂ t' 1 = ∂ T ||T' ||

( I − 13 I ⊗ I − t' ⊗ t' )

(1.243)

1.14 Partial Differential Calculi

49

cos3θ ≡ 6t r t' 3

(1.244)

∂ cos3θ = ∂ 6 t'pq t'qr t'rp = 3 δ δ t t = 3 t t , 6 ip jq qr 6 'ir 'rj ' 'rp ∂t'ij ∂t'ij ∂ cos3θ = 3 6 t' 2 ∂t'

(1.245)

∂ cos3θ ∂ cos3θ ∂t'rs = ∂t'rs ∂ Tij ∂ Tij

= 3 6 t'rt tts'

1 Tpq ' Tpq '

=3 6

{12 (δ irδ js + δ isδ jr ) − 13 δ ijδ rs − t'ijt'rs}

1 Tpq ' Tpq '

∂ cos 3θ 1 =− ∂T ||T' || df (T) =

(tir' trj' − 1 trt' ttr' δ ij − trs' t'st ttr' t'ij ) , 3

(

2 2 6 ||t' || I + 3cos 3θ t' − 3 6 t' )

∂f (T) ∂f (T) d TT) dTrs =tr ( ∂T ∂ Trs

(1.246)

(1.247)

Differentiating Tir−1Trs = δ is , one has 1 ∂ (Tir−1Trs ) ∂Tir−1 ∂Tir− −1 = T T T −1 + Tir−1δ rk δ slTsj−1 = 0 rs + Tir δ rk δ sl = 0 → ∂Tkl ∂Tkl rs sj ∂Tkl

which leads to

∂Ti j−1 = − 1 (Tik−1T jl−1 + Til−1T jk−1) 2 ∂Tkl Differentiating Eq. (1.145), it holds that

Ⅲ

Ⅰ Ⅱ

∂ = T 2 − T+ I ∂T

50

1 Tensor Analysis

Then, noting Eq. (1.176), one has

Ⅲ

Ⅰ Ⅱ ⅢT

∂detT = ∂ = T2 − T+ I = ∂T ∂T

−1

= (detT) T−1

which was derived also in Eq. (1.105). For asymmetric tensors, the tensor δ ijδ kl has to be used instead of the symmetric identity tensor (δ ik δ jl + δ il δ jk ) /2 in the above equations.

1.15 Time Derivatives Let the tensor T , describing a physical quantity, be denoted by T(x, t ) in the spatial description fixed in a space and by T(X, t ) in the material description moving with a material, where t is a time. The time derivative in the former, i.e.

∂T(x, t ) ∂t

(1.248)

describes the rate of the tensor T in the fixed point in the space and then it is called the spatial-time (or local) derivative. In many cases of fluid mechanics, a mechanical variation of individual particle of fluid is not important since deformation of fluid from the initial state does not influence the movement of fluid and thus the spatial-time derivative is often adopted. On the other hand, the time derivative in the material description, i.e.

∂T( X, t ) ∂t

(1.249)

describes the rate of the tensor T in the particular particle of material and thus is called the material-time derivative. It is denoted by the symbol •

T≡

∂T( X, t ) ∂T( X, t ) or DT ≡ Dt ∂t ∂t

(1.250)

In solid mechanics, the rate of deformation in individual particles of a solid is important and thus the material-time derivative is used usually. Because of T( X( x, t ), t ) = T( x, t ) the material-time derivative in Eq. (1.250) and the spatial-time derivative in Eq. (1.248) is related as follows: •

T≡

∂T(x, t ) ∂T(x, t ) v + ∂t ∂x

(1.251)

where v ≡ ∂x/∂t is the velocity vector of material particle. The first term in the right-hand side signifies the non-steady (or local time derivative) term describing

1.16 Differentiation and Integration in Field

51

the change with the elapse of time and the second term signifies the steady (or convective) term describing the change attributable to the movement of material, which results from the existence of a gradient in the mechanical state. Rate-type constitutive equations describing the irreversible deformation, e.g. the viscoelastic, the elastoplastic and the viscoplastic deformation, must be described by the material-time derivative pursuing a material particle because it describes the relation of physical quantities in a material particle. Further, as described in 4.3, the corotational derivative based on the rate of mechanical state observed by the coordinate system rotating with a material must be used in general.

1.16 Differentiation and Integration in Field Scalar s , vector v , and tensor T are called the scalar field, the vector field, and the tensor field, respectively when they are functions of the position vector x . Their differentiation and integration in fields are shown below, in which the following operator, called the nabra or Hamilton operator, is often used.

∇ ≡ ∂ er = ∂ ∂xr ∂x

(1.252)

1) Gradient

∂s Scalar field: grad s = ∇ s = x e r ∂

(1.253)

r

Vector field:

⎧

　⎪ v ⊗ ∇ = vi ei ⊗ ∂∂x j e j = ∂∂xvij ei ⊗ e j :

rear (right) form ⎪ gradv = ⎨ ⎪∇ ⊗ v = ∂ ei ⊗ v j e j = ∂v j ei ⊗ e j : front 　 ( left) form ∂xi ∂ xi ⎪⎩

(1.254)

Second-order tensor field:

⎧

　⎪ T ⊗ ∇ = Tij ei ⊗ e j ⊗ ∂∂x e k = ∂∂Txijk ei ⊗ e j ⊗ e k :

rear (right) form k ⎪ grad T = ⎨ ⎪∇ ⊗ T = ∂ ei ⊗ T e ⊗ e = ∂ Tjk ei ⊗ e ⊗ e : front 　 ( left) form j jk j k k　 ⎪ ∂xi ∂xi ⎩ (1.255)

2) Divergence Vector field: div v = ∇ • v 　( = v • ∇ ) = vi ei

•

v ∂ e =∂ i ∂x j j ∂xi

(1.256)

52

1 Tensor Analysis

Second-order tensor field:

⎧

　 ( right) form 　⎪ T • ∇ = Tij e i ⊗ e j • ∂∂x e k = ∂∂Txirr ei : rear

k ⎪ (1.257) div T = ⎨ ri T ∂ ⎪∇ • T = e • e e = ∂ e : front 　 ( left) form ∂xi i Tjk j ⊗ k ∂x r i　 ⎪⎩

3) Rotation (or curl) Vector field:

⎧

　 ( right) form 　⎪ v × ∇ = viei × ∂∂x j e j = ε ijk ∂∂xvij ek : rear

⎪ rotv = ⎨ ⎪∇ × v = ∂ ei × v j e j = ε ijk ∂v j ε ijk ek : front 　 ( left) form ∂xi ∂ xi ⎪⎩

(1.258)

noting Eq. (1.36). Second-order tensor field:

⎧

　⎪ T × ∇ = Tij ei ⊗ e j × ∂∂x e k

k ⎪ ⎪ ∂Tij ∂Tij 　 ) form ⎪ = x ei ⊗ (e j × e k ) = ε jkr x e i ⊗ e r : rear ( right ∂ k ∂ k ⎪ rot T = ⎨ ⎪∇ × T = ∂ ei × Tjk ej ⊗ ek ∂xi ⎪ ⎪ ∂ Tjk ∂Tjk ⎪ 　 ( left) form = x (ei × ej ) ⊗ ek　 = ε ijr x e r ⊗ e k : front ⎪ ∂ i ∂ i ⎩

(1.259) The symbol ∇ is regarded as a vector, and the scalar product of itself, i.e. 2 Δ ≡ ∇ 2 ≡ ∇ • ∇ = ∂x e r • ∂ e s = x∂ x r ∂ r ∂ ∂ r ∂ xs

(1.260)

has the meaning of ∇ 2 ( ) ≡ div(grad( ) ) . The symbol Δ is called the Laplacian or Laplace operator, which is often used for scalar or vector fields as 2s Δ s = x∂ x ∂ r∂ r

(1.261)

1.16 Differentiation and Integration in Field

53

2 vs Δv = ∂x x e s ∂ r∂ r

(1.262)

The following relations hold between the above-mentioned operators.

grad( sv ) = v ⊗ grads + sgradv,

½ ° div( sv ) = sdivv + v • grads, ° ° div(u × v ) = v • rotu − u • rotv, ° rot(u × v ) = (gradu) v − (gradv )u + (divv )u − (divu) v, ¾ (1.263) grad(u • v ) = (gradv )u + (gradu) v + u × rotv + v × rotu, ° ° ° div( sT) = TT grads + sdivT, ° div(Tv ) = v • divT + tr(Tgradv ) °¿ 4) Gauss’ divergence theorem Consider the physical quantity T ( x ) in the zone surrounded by a smooth surface inside a material. Then, suppose the slender prism cut by the four planes perpendicular to the x2 -axis and x3 -axis in infinitesimal intervals from a zone inside the material. The following equation holds for the prism.

∫v ∂∂xT dv = ∫v ∂∂xT dx dx dx 1

1

1

2

3

x+ = [T ]x1− dx2 dx3 = (T + − T − ) dx2 dx3 (1.264) 1

where ( )+ and ( )− designate the values of physical quantity at the maximum and the minimum x1 -coordinates, respectively. The neighborhood of the surface cut by the prism is magnified in Fig. 1.7. Consider the infinitesimal rectangular surface PQRS of the prism exposed at the surface in the maximum x1 -coordinate and the infinitesimal rectangular section PQ∗R ∗S∗ cut by the plane passing through the point P and perpendicular to the → → x1 -axis by the prism. Then, denoting QQ ∗ = dxQ , SS∗ = dx S , the vectors PQ, PS

are given by

→ → PQ = dx2e 2 + dxQe1 , PS = dx3e3 + dxSe1 and thus it holds that

→ → n + da + = PQ× PS = (dx2e 2 + dxQe1 ) × (dx3e3 + dxSe1 )

(1.265)

54

1 Tensor Analysis

R∗

S∗∗ dx ss S dx e3

R

nn++

dx33 dx

Q ∗∗ Q

e2

dx QQ dx Q

0 P

e1

Fig. 1.7 Infinitesimal square pillar cut from a zone in material

= dx2 dx3e 2 × e3 + dxQ dx3e1 × e3 + dxSdx2e 2 × e1 = dx2 dx3e1 − dxQ dx3e 2 − dxS dx2e3

(1.266)

Comparing the components in the base e1 on the both sides in Eq. (1.266), one has

n1+ da + = dx2 dx3

(1.267)

In a similar manner for the surface of the prism exposed on surface in the minimum x1 -coordinate, one has

n1− da − = − dx2 dx3

(1.268)

The general expression of projected area is given in Appendix 1. Adopting Eqs. (1.267) and (1.268) in Eq. (1.264), it holds for the prism that

∫v ∂∂Tx dv = T +n+ da + − T − (−n − da − ) = T +n+ da + + T − n− da − 1

1

1

1

(1.269)

1

Then, the following equation holds for the whole zone.

∫v ∂∂Tx dv = ∫a T n da

(1.270)

1

1

In a similar manner also for the x2 -direction and x3 -direction, the following Gauss’ divergence theorem holds.

∫v ∂∂Txi dv = ∫aT ni da

(1.271)

1.16 Differentiation and Integration in Field

The following equations for the scalar from Eq. (1.271).

55

s , the vector v and the tensor T

hold

∫v ∂∂xi dv = ∫a s ni da, ∫v ∇s d v = ∫a s n da

(1.272)

∫v ∂∂ xii dv = ∫a vi ni da, ∫v ∇ • vdv = ∫a v • n da

(1.273)

∂ Tij

(1.274)

s

v

∫v ∂ xj dv = ∫aTij ni da, ∫vT • ∇ dv = ∫a Tn da

5) Material-time derivative of volume integration Supposing that the zone of material occupying the volume v at the current moment (t = t ) changes to occupy the volume v + δ v after the infinitesimal time

(t = t + δ t ) , the material-time derivative of the volume integration

∫v T (x, t )d v of

the physical quantity T (x, t ) involved in the volume is defined by the following equation.

1{ (∫v T dv)• = δlim ∫v+δ v T (x, t + δ t)dv − ∫v T (x, t)dv} t →0

δt

= lim 1 [ ∫ {T ( x, t + δ t ) − T (x , t )}dv + ∫ T (x, t + δ t ) dv] δv δ t →0 δ t v (1.275) The integration of the first term in the right-hand side in Eq. (1.275) is transformed as

T (x, t ) lim 1 ∫ {T (x, t + δ t) − T (x, t )}dv = ∫ ∂ dv v v δ t →0 δ t ∂t

(1.276)

On the other hand, the second term in Eq. (1.275) describes the influence caused by the change of volume during the infinite time. Here, the increment of volume, δ v , is given by subtracting the volume going out the boundary of the zone from the volume going into the boundary, which is the sum of dv (=v • ndaδ t ) over the whole boundary surface (Fig. 1.8). Therefore, substituting the Gauss’ divergence theorem (1.271) and ignoring the second-order infinitesimal quantity, the integration of the second term in the right-hand side of Eq. (1.275) is given by

lim 1

∫

δ t →0 δ t δ v

T ( x, t + δ t ) dv ≅ lim 1 ∫ T (x, t ) dv δ t →0 δ t δ v

∂T ( x, t )vr = lim 1 ∫ T ( x, t )vr nr daδ t = ∫ T (x, t )vr nr da = ∫ dv a v δ t →0 δ t a ∂ xr

56

1 Tensor Analysis

vδ t

a

a

n

da

da

vδ t v

n

v +δv

Fig. 1.8 Translation of a zone in material

The sum of the first term in the right-hand side in this equation and the Eq. (1.276) is equal to the material-time derivative of T (x, t ) because of Eq. (1.251) and thus Eq. (1.275) is given by

(∫v T (x, t ) dv)• = ∫v{T (x, t ) + T (x, t ) ∂∂vxrr}dv = ∫v{T (x, t ) + T (x, t )divv}dv •

•

(1.277) which is called the Reynolds’ transportation theorem. Equation (1.277) can be obtained also by the following manner.

(∫v T (x, t ) dv)• = (∫V T ( X, t ) JdV )• = ∫v (T• ( X, t ) J + T ( X, t ) J• ) dV ∂v = ∫ {T (x, t ) + T (x, t ) r }dv v ∂ xr •

where V is the initial volume of zone and it is set that J ≡ dv/dV . Here, it holds • that J = J (∂vr / ∂ xr ) as will be described in 2.5. For the physical quantity T kept constant in a volume element, Eq. (1.277) leads to •

∫v (T (x, t ) + T (x, t ) divv)dv = 0

(1.278)

The local (weak) form of Eq. (1.278) is given as •

T (x, t ) + T (x, t ) divv = 0

(1.279)

Chapter 2 Motion and Strain (Rate)

2

Motion and Strain (Rate)

The tensor analysis providing the mathematical foundation for the continuum mechanics is described in Chapter 1. Basic concepts and quantities for continuum mechanics will be studied in the three chapters up to Chapter 4. The description of motion is required in the continuum mechanics. Various measures for the motion and strain (rate) of solids are adopted for the description of reversible and irreversible deformation of materials. Main measures among them will be explained in this chapter.

2.1 Motion and Deformation General, basic quantities and manners for describing the motion and deformation will be shown in this section.

2.1.1 Material, Spatial and Relative Descriptions A material body is assembly of material particles (or material elements). The positions of all material particles in a laboratory (Euclidean space) is referred to as the configuration of the material body. Here, the configurations in the initial time t = 0 and the current time t are called the initial configuration and the current configuration, respectively, and the position vectors of material particle in the initial and the current configurations are designated by X and x(t ) , respectively. Here, X is fixed and thus it can be regarded as a label of each material particle. At fixed time the unique relation exists between X and X provided that the material does not overlap or separate. The mapping may be symbolically described as x = χ ( X, t ), X = χ −1 (x, t )

(2.1)

As discussed in 1.15, a physical quantity, say T , representing the state of the body changes with the position of material particle, i.e., with the change of the body-configuration and the time. The special reference configuration is usually selected to identify the material particles of the body and it is called the reference configuration. Here, we adopt the initial configuration to be the reference configuration. When X is regarded as an independent variable, the field of physical quantity is described by T(X, t ) . This type of description of material state and properties is called the Lagrangian (or material) description (cf. Section 1.5). On

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 57–99. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

58

2 Motion and Strain (Rate)

the other hand, when we adopt current position of a material particle as independent variable, then the field of physical quantity in the domain of current configuration is described by T(x, t ) . This way of description is called the Eulerian description or spatial description, as already mentioned in 1.5. The former follows the variation of the state of a material particle, whereas the latter observes the variation of physical quantity at a fixed point of a laboratory. A configuration at a particular time t = τ , i.e. x(τ ) can be described from Eq. (2.1) as follows:

x(τ ) = χ ( X,τ ), X = χ −1 (x(τ ),τ )

(2.2)

The state of physical quantity in a body may be described in terms of x(τ ) as −1 T(χ (x(τ ),τ ), t ) = τ T{x(τ ), t}

(2.3)

Here, the set of x(τ ) is called the relative configuration and the description presented in Eq. (2.3) is called the relative description. Specifically, tT{x(t ), t} (τ =t ) leading to T( x, t ) is called the updated Lagrangian description. In this case the current configuration is regarded as the reference configuration and, for the sake of clarity the type of description T( X, t ) will be called the total Lagrangian description. The fact that a material does not overlap or separate by the deformation requires the existence of the one-to-one correspondence between X and x ( x is uniquely determined for X and vice versa) so that x1 ( X 1 X 2 X 3 ), x2 ( X1 , X 2 , X 3 ) and x3 ( X 1 X 2 X 3 ) must be mutually independent. Now, assume that x1 , x2 , x3 are not mutually independent. Then, there exists a function f such that

f ( x1 ( X 1 X 2 X 3 ), x2 ( X 1 , X 2 , X 3 ), x3 ( X 1 X 2 X 3 )) = 0

(2.4)

from which one has

⎫ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 =0 ⎪ + + ∂x1 ∂X 1 ∂x2 ∂X 1 ∂x3 ∂X 1 ⎪ ⎪ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 = 0⎬ + + ∂x1 ∂X 2 ∂x2 ∂X 2 ∂x3 ∂X 2 ⎪ ⎪ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 = 0⎪ + + ∂x1 ∂X 3 ∂x2 ∂X 3 ∂x3 ∂X 3 ⎪⎭

(2.5)

2.1 Motion and Deformation

59

Eliminating ∂f / ∂x1 , ∂f / ∂x2 , ∂f / ∂x3 from Eq. (2.5), one gets

J ≡ det (

∂xi ∂X J

) = ε IJK ∂∂Xx I 1

∂x2 ∂x3 =0 ∂X J ∂X K

(2.6)

on account of Eq. (1.10), where J is called the functional determinant or Jacobian. This result shows that functions x1 , x2 , x3 are not independent. In order that they are independent, it must hold that

∂x det ( i ∂X J

)≠0

(2.7)

The transformation between x and X is called the admissible transformation, if f1 , f 2 , f3 in x1 = f1 ( X1 X 2 X 3 ), x2 = f 2 ( X1, X 2 , X 3 ) , x3 = f 3 ( X 1 X 2 X 3 ) are singlevalued and continuous functions and the Jacobian is not zero at all points. Further, if the Jacobian is positive at all points, a right-hand coordinate system is transformed to other right-hand one, and it is called the positive transformation. Inversely, if the Jacobian is negative at all points, a right-hand coordinate system is transformed to a left-hand one, and it is called the negative transformation. Admissible and positive transformation is assumed throughout this book.

2.1.2 Deformation Gradient and Deformation Tensors At the initial state of deformation ( t = 0), consider the material particle, the position vector of which is X , and let the position vector of the adjacent material point be X + d X . Furthermore, consider the current state ( t = t ) in which these material particles moved to the space points having position vectors x and x + dx . The line elements before and after the deformation are described as

dX = dX A e A , dx(t ) = dxi (t ) ei (t )

(2.8)

where the different bases are assumed for the initial and current configurations so that the observer can move with a deformation. Here, define the deformation gradient

(t ) F(t ) ≡ ∂x , ∂X

FiA (t )ei ⊗ eA ≡

∂xi (t ) e ⊗ eA ≡ x i , A (t )ei ⊗ eA ∂X A i

(2.9)

Then, dx(t ) is described by dX as follows:

dx(t ) = F(t )dX, dxi (t ) = FiA (t ) dX A ,

dxi (t )ei (t ) = FiAei (t ) ⊗ e A dXBe B = FiA dXAei (t )

(2.10)

60

2 Motion and Strain (Rate)

Equation (2.7) is written in terms of the deformation gradient as

J = det F ≠ 0 Therefore, the inverse (∂x/∂X )(∂X/∂x ) = I as

F −1 = ∂X , ∂x

tensor

F −1

(F −1 ) Ai eA ⊗ ei =

(2.11)

exists,

which

is

derived

∂X A e ⊗ ei ≡ X A, i (t )eA ⊗ ei ∂ xi A

from

(2.12)

Here, consider the unit cubic cell (a parallelepiped) whose sides at initial configuration are given by the triad {eA } . It deforms to the cell whose sides are formed by the triad {eA } while eA are not unit vectors in general but are given from Eq. (2.10) as follows:

eA = FeA (FiA = ei • FeA = ei • eA )

(2.13)

Applying the polar decomposition in 1.10 to the deformation gradient F , we have F = RU = VR ( FiA = RiRU RA = Vir RrA )

(2.14)

where U = (FT F)1/2 (U 2 = FT F) (UT = U)

(2.15)

V = (FFT )1/ 2 (V 2 =FFT ) (VT =V)

(2.16)

R = FU −1 = F (FT F ) −1/ 2 , R = V −1F = (FFT ) −1/ 2 F (det R = 1) (2.17) V = RURT , U=RT VR

(2.18)

Since Eq. (2.18) holds and R is the orthogonal tensor, U and V are the similar tensors as discussed in 1.5.2. Therefore, they possesses the principal values

λ α (α = 1, 2, 3) . Denoting the bases for the principal directions of U and V by {Nα (t )} and {n (α ) (t )} , respectively, it can be written as 3

3

α =1

α =1

U = ∑ λα N (α ) (t ) ⊗ N (α ) (t ) , V = ∑ λα n (α ) (t ) ⊗ n (α ) (t )

(2.19)

where the relation of N(α ) (t ) and n(α ) (t ) is given from Eq. (1.189) as follows:

n(α ) (t ) = R (t )N(α ) (t ), N(α ) (t ) = RT (t )n(α ) (t )

(2.20)

2.1 Motion and Deformation

61

with R(t ) = n(α ) (t ) ⊗ N(α ) (t )

(2.21)

N(α ) (t ) and n (α ) (t ) are called the Lagrangian triad and the Eulerian triad,

respectively. Substituting Eqs. (2.19) and (2.21) into Eq. (2.14), F is described by 3

F(t ) = ∑ λ α (t )n (α ) (t ) ⊗ N (α ) (t )

(2.22)

α =1

Let the mechanical meanings of U, V and R be examined below. The variation of infinitesimal line-element is given by the polar decomposition F = RU noting UN (α ) = λα N (α ) (no sum) as follows: 3

3

α =1

α =1

dx = FdX = RUdX = RU ∑ dXα N (α ) = R ∑ λα dXα N (α )

(2.23)

Equation (2.23) means that the infinitesimal line-elements dXα N (α ) (no sum) in the principal directions N (α ) are first stretched λα times to λα dXα N (α ) (no sum) and then undergoes the rotation R as shown in Fig. 2.1. On the other hand, the change of the infinitesimal line-element by the polar decomposition VR is described as 3

3

3

α =1

α =1

α =1

dx = VRdX = V ∑ R dXα N (α ) = V ∑ dXα n (α ) = λ α ∑ dXα n (α )

(2.24)

Equation (2.24) means that the infinitesimal line-elements dXα N (α ) (no sum) in the principal directions N (α ) first becomes dXα n (α ) (no sum) by rotation R and then are stretched λα times to λα dXα n (α ) (no sum) , noting Vn(α ) = λα n(α ) (see Fig. 2.1). As described above, U, V designates the deformation and R the rotation. λα is called the principal stretch and U and V are called the right and left stretch tensor, respectively. Letting R L and RE designate the rotations of the Lagrangian triad {N(α ) } and the Eulerian triad {n (α ) } , respectively, from the fixed base {eα } (α = 1, 2,3) , they are given by 3

3

α =1

α =1

RL ≡ ∑ N (α ) ⊗ eα , RE ≡ ∑ n(α ) ⊗ eα

where the following relations hold.

(2.25)

62

2 Motion and Strain (Rate)

Rd

(2

n

)

N

(2 )

(2

e2

X X

)

(1

n (1

R

)

n

R

(1 )

N

X X Fd =F x d

=R

Ud

X

=V

R

dX

)

n

R

E

0 0

R

X2

L

e1 X dX X2 dX 2 X X λ2 d

(2 )

N

N

(2 )

N N dX 1 1 λ 1 dX

e2 (1

n

X1

(1 )

0

) (2

X UddX

n

1 1

)

R (1 )

N e1

R

(2

E

)

n

E

U

λλ 2

R (2 )

N

(1

n

)

λλ 1

R

e2

(2 )

N (11))

1

N

1

e1

λ2

RL

(1 )

U

Fig. 2.1 Polar decomposition of the deformation gradient

N N

λ1

2.1 Motion and Deformation

63

N (α ) = RL eα , n(α ) = RE eα RE = RRL

(2.26) (2.27)

Considering the particle P and the adjacent particles P' and P'' , we designate their position vectors before and after the deformation by X, X + dX, X + δ X and x, x + dx, x + δ x , respectively. Then, noting (1.100), one has

dx • δ x = FdX • Fδ X = FT FdX • δ X = CdX • δ X

(2.28)

−

dX • δ X = F −1dx • F −1δ x = F T F −1dx • δ x = ( F F T ) −1 d x • δ x = b −1 d x • δ x

(2.29)

C ≡ FT F = U 2 (C = CT ), C AB = FkA FkB

(2.30)

where

and b ≡ FFT = V 2 (= RCRT ) ( b = bT ) , bij = FiA F jA

(2.31)

are the metric tensors describing how the scalar product of two line-element vectors passing through a material point is influenced by a deformation. They are called the right and left Cauchy-Green deformation tensor. In accordance with Eq. (2.19) they are described by 3

3

α =1

α =1

C = ∑ λα2 N(α ) ⊗ N (α ), b = ∑ λα2 n(α ) ⊗ n (α ) , The principal values equation

λa

are obtained by the solutions of the characteristic

Ⅰ Ⅱ λ +Ⅲ

λ 3 − cλ 2 − where

Ⅰ ≡ trC, Ⅱ ≡ 12 (trC − trC ), Ⅲ c

c

(2.32)

2

c

c

c

=0

(2.33)

1 1 3 3 2 1 ≡ 6 tr C − 2 tr C t r C + 3 t rC (2.34)

The principal values and directions are calculated by the method described in 1.5.2. Using the relative description (2.3),the relative deformation gradient on the reference configuration x(τ ) is defined as τ F{x(τ ), t} =

∂ τ χ{x(τ ), t} ∂x(τ )

(2.35)

64

2 Motion and Strain (Rate)

which is hereinafter shown in the abbreviated notation as follows: τ F (t ) ≡ τ F{x(τ ), t} τ F (t )

(2.36)

is related to the deformation gradient F(t )(≡ 0F(t )) as

F (t ) =

∂x(t ) ∂x(t ) ∂x(τ ) = = τ F(t )F (τ ) ∂X ∂x(τ ) ∂X

(2.37)

and is further expressed in the polar decomposition as τ F (t ) = τ R (t ) τ U (t ) = τV (t )τ R (t )

(2.38)

where τ C(t ), τ b(t ) defined by 2 τ C(t ) = ( τ F (t )) T τ F (t ) = τ U (t ) ½ ° 2

(2.39)

¾

τ b(t ) = τ F (t ) (τ F (t ))T = τV (t ) ° ¿

are called the relative right, left Cauchy-Green tensors.

2.2 Strain Tensor Taking the subtraction of Eqs. (2.28) and (2.29), one has

dx • δ x − dX • δ X = 2EdX • δ X ( = 2E AB dX Aδ X B ) (2.40)

= 2 edx • δ x ( = 2eij dxi δ x j ) where ⎫ ⎪ ⎪ ∂x k ∂x k ⎪ 1 1 E AB ≡ ( FAk FkB − δ AB ) = ) − δ AB ⎪ 2 2 ∂X A ∂X B ⎪ ⎬ T e ≡ 1 (I − b−1 ) = 1 (I − F −T F −1 ) = 1 I − ∂X ∂X ⎪ 2 2 2 ∂x ∂x ⎪ ⎪ ∂X ∂XK ) ⎪ eij ≡ 1 {δ ij − (F −1 ) Ki (F −1 ) K j} = 1 (δ ij − K 2 2 ∂xi ∂xj ⎪⎭

1 1 T 1 E ≡ 2 (C − I ) = 2 (F F − I) = 2

{( ∂∂Xx )T ( ∂∂Xx ) − I}

(

{ ( )

}

(2.41)

Applying the quotient law described in 1.3.2 to Eq. (2.40), it is confirmed that E and e are the second-order tensors.

2.2 Strain Tensor

65

If a deformation is not induced, the triangle PP' P'' has the same shape as in the initial state and thus the scalar quantities in the left-hand side in Eq. (2.40) are zero so that E and e are independent of rigid-body rotation. Conversely, if E ≠ 0, e ≠ 0 , the scalar quantities in the left-hand side in Eq. (2.40) are not zero so that the shape of the triangle is not same as in the initial state. Therefore, E and e are the quantities describing the deformation independent of rigid-body rotation and called the Green (or Lagrangian) strain tensor and the Almansi (or Eulerian) strain tensor, respectively. Using the displacement vector

u = x − X = u ie i

(2.42)

they are expressed by 1 ∂u + ( ∂u )T + ( ∂u )T ( ∂u )} , E = 1 ( ∂u A + ∂u B + ∂ uK ∂u K ) ⎫ E = 2{ AB 2 ∂X B ∂ X A ∂X A ∂X B ⎪ ∂X ∂X ∂X ∂X ⎪ ⎬ ∂ u 1 ∂u ∂u ∂u k j u u T u T u ⎪ e = 1 {∂ + ( ∂ ) − ( ∂ ) ( ∂ )} , e ij = ( i + − k ) 2 ∂x ∂x ∂x ∂x ⎪⎭ 2 ∂x j ∂xi ∂xi ∂x j (2.43) The following relation exists between them.

E = FT eF,

E AB = FiA FjB eij

(2.44)

Now, limiting to the infinitesimal deformation and rotation, dx ≅ d X holds and thus the third term becomes the second-order infinitesimal quantity in Eq. (2.43) so that the difference between E and e can be ignored. Then, denoting E or e by ε , it can be written that T⎫ 1⎧ ε ≡ ( ∂u )S = ⎨ ∂u + ( ∂u ) ⎬ , ∂X 2 ⎩ ∂X ∂X ⎭

∂ u A ∂ uB + ) 2 ∂XB ∂XA

ε AB ≡ 1 (

(2.45)

or T⎫ 1⎧ ε ≡ ( ∂u )S = ⎨ ∂u + ( ∂u ) ⎬ , x x x 2 ⎩∂ ∂ ∂ ⎭

∂u ∂u j ε ij ≡ 1 ( i + ) 2 ∂xj ∂xi

(2.46)

ε is called the infinitesimal strain tensor. ε is not an exact measure to describe a deformation for finite deformation and rotation since it does not describe the relation of the infinitesimal line-elements dX and dx directly, and thus it possesses various impertinence as will be described in 2.3.

66

2 Motion and Strain (Rate)

Consider the same infinitesimal line-element for PP' and PP'' , i.e. dX = δ X, dx = δ x and denoting its direction vector by N (||N||= 1) , it holds from Eq. (2.40) that

||dx||2 − ||dX||2 = 2EN • N ||dX||2

(2.47)

Selecting the X 1 -axis for this line-element, ( N1 , N2 , N3 ) = (1, 0, 0) holds and thus we have

|| dx||2 − || dX||2 E11 = 1 2 || dX||2

(= 12 ||dx||||d−X||||dX|| ( ||||ddXx|||| + 1))

(2.48)

from which the ratio of the line-elements before and after the deformation is given by

||dx|| = 1 + 2 E11 || dX||

(2.49)

In the case that the variation of the length of the line-element is infinitesimal (|| dx|| / || dX|| ≅ 1) , Eq. (2.48) becomes

E11 ≅ ε11 =

|| dx|| − || dX|| || dX||

(2.50)

Then, E11 becomes to describe the rate of elongation coinciding with the normal strain in the infinitesimal strain ε. On the other hand, denoting the direction vectors of the two distinct infinitesimal line-element PP' and PP'' as N' and N'' , respectively, and the angles contained by them as θ , it holds from Eq. (2.40) that

|| dx||||δ x||cosθ − ||dX|||| δX|| cosθ 0 = 2EN' • N'' ||dX|||| δX|| (2.51) i.e.

|| dx|| ||δ x|| cosθ − cosθ 0 = 2EN' • N'' = 2 E ijN'i N''j || dX|| || δX||

(2.52)

where θ 0 is the initial value of θ . Here, assuming that the infinitesimal line-elements PP' and PP'' were mutually perpendicular before a deformation ( θ0 = π /2 ), and taking the X 1 - and X 2 -axes to their directions, i.e.

( N1' , N'2 , N'3 ) = (1, 0, 0), ( N1'' , N''2 , N''3 ) = (0, 1, 0) , it holds that

||dx|| ||δ x|| ||dx|| ||δ x|| E12 (= 1 cos θ ) = 1 sin(π /2 − θ ) 2 ||dX|| || δX|| 2 ||dX|| || δX||

(2.53)

2.2 Strain Tensor

67

which describes the half of decrease in the sine of angle contained by the two line-elements which were mutually perpendicular before deformation when the changes in lengths of these line-elements are infinitesimal ( || dx|| / || dX|| ≅1, ||δ x|| / || δX|| ≅ 1 ). Furthermore, when the change in the angle formed by these line-elements is infinitesimal ( θ ≅ π /2 ), one has

E12 ≅ ε12 = (1/2) tan(π /2 − θ ) ≅ (π /2 − θ ) / 2

(2.54)

Consequently, E12 describes half of the decrease in the angle contained by the two line-elements which were perpendicular before deformation. In addition to the Lagrangian and Eulerian strain tensors defined above, we can define various strain tensors in terms of U or V , fulfilling the condition that they are zero when U = V = I as follows (Seth, 1964; Hill, 1968):

⎫ 1 m 1 m m (U − I), m (V − I) for m ≠ 0 ⎪⎬ lnV ln U, for m = 0 ⎪⎭

(2.55)

where m is the integer (positive or negative). The Green strain tensor is obtained by choosing m = 2 in the equation of U and the Almansi strain tensor is obtained by choosing m = −2 in the equation of V in Eq. (2.55). The Biot strain tensor U − I (Biot, 1965) is given by m = 1 . The strain tensors in Eq. (2.55) are coaxial with U or V and their principal values are given by

⎧1 m − ⎪ (λα 1) for m ≠ 0 f (λ α ) = ⎨ m ⎪⎩ ln λα for m = 0

(2.56)

The function f (λα ) fulfills

f (1) = 0, f ' (1) = 1

(2.57)

and

f ' (s) > 0

(2.58)

where s is an arbitrary positive scalar quantity. The function f (λ α ) is shown in Fig. 2.2. (Note) it holds for m = 0 Eq. (2.56) that 1 (λm − 1) = lim exp(m ln λα ) − 1 = lim exp(m ln λα ) ln λα = lim m ln λα α m m →0 m →0 1

m→ 0

68

2 Motion and Strain (Rate)

m=2

f (λα ) 1.5

m =1

1 m=0 m = −1

0.5 0

0.5

1

m=2

2

λα

−0.5

−1

m = 0 m = −1 Fig. 2.2 Function of general principal strain measures

Further, adopt the second-order tensor function f (U) which is coaxial with the right stretch tensor U and has the principal values f (λα) . Therefore, we can define the general strain tensor in the spectral representation as follows: 3

f (U) = ∑ f (λα )N(α ) ⊗ N (α )

(2.59)

α =1

In addition, for the left stretch tensor V , we can define the following strain tensor. 3

3

α =1

α =1

f (V ) = ∑ f (λα )n (α ) ⊗ n (α ) = ∑ f (λα )RN (α ) ⊗ RN (α ) = Rf (U)RT (2.60) The relation (Tu ⊗ Tv)ij = Tir ur Tjs us = Tir ur usTjs = (T(u ⊗ v )TT )ij leading to Tu ⊗ Tv = T(u ⊗ v )TT is used in the derivation of the last side in Eq. (2.60).

2.2 Strain Tensor

69

In the particular case of m = 0 , noting Uα ≥ 0, Vα ≥ 0 , f (U) and f (V) defined by the following equation are called the Hencky strain tensor. 3 ½ Right - Hencky strain tensor : Ȝr = ¦ λα N (α ) ⊗ N (α ) ≡ lnU = 1 lnC ° ° 2 α =1 ¾ 3 ° Left - Hencky strain tensor : Ȝl = ¦ λα n (α ) ⊗ n (α ) ≡ lnV = 1 lnb °¿ 2 α =1

(2.61) where

λ α ≡ lnUα = 1 lnCα = lnVα = 1 lnbα (Uα = Vα , Cα = bα ) 2 2

(2.62)

which are mutually related as follows.

λr = RT λl R

(2.63)

When the principal directions of C and b are fixed, choosing the coordinate axes to these directions FiA = 0 (i ≠ A) ), the following equations hold.

λα = ln ∂xα (no sum) ∂Xα

λ α = λ α = ( ∂xα •

•

∂Xα

•

)

∂x /( ∂Xαα

(2.64)

•

) = ∂∂xxαα = Dαα (no sum)

(2.65)

where λα in Eq. (2.64) is the logarithmic strain and Dαα (no sum) is the normal component in the strain rate tensor defined in the next section. It holds from Eq. (2.64) that 3 3 3 t rλ r = ∑ λα = 1 ∑ ln Cα =∑ lnU = ln(U U U ) ⎫ I II III α ⎪ 2 α =1 ⎪ α =1 α =1 ⎬ 3 3 3 ⎪ t rλl = ∑ λα = 1 ∑ ln b α =∑ lnV = ln(V V V ) α I II III 2 ⎪⎭ α =1 α =1 α =1

(2.66)

which is identical to the logarithmic volumetric strain, i.e. 3 ∂xα v trλ r = trλl = ∑ ln ∂X = ln J = ln V = ε v α

α =1

(2.67)

70

2 Motion and Strain (Rate)

2.3 Strain Rate and Spin Tensors The idealized deformation process in which the deformation is uniquely determined by the state of stress independent of the loading path is called the elastic deformation process. To describe it, it is required to introduce the strain tensor describing the deformation from the initial state and relate it to the stress. Here, since the superposition rule does not hold in the strain tensor, the null stress state is chosen as the reference state of strain. On the other hand, the deformation is not determined uniquely by the state of stress depending on the loading path and thus it cannot be related to the stress in the irreversible deformation process, e.g. the viscoelastic, the plastic and the viscoplastic loading processes. Therefore, it is obligatory to relate the infinitesimal changes of stress and deformation and to integrate them along the loading path in order to know the current states of stress and deformation. Here, introduce the velocity gradient tensor defined as

∂v L ≡ ∂v , Lij ≡ i ≡ ∂ j vi ∂x ∂x j

(2.68)

•

Noting F• = ∂ x• / ∂X = ∂v / ∂X (dv = FdX ) and the chain rule of derivative, Eq. (2.68) can be rewritten as • • ∂v ∂X A L = ∂v ∂X = FF -1 (F = LF ), Lij = i ∂X ∂x ∂X A ∂x j

(2.69)

Now, differentiating Eq. (2.36) and choosing the current state as the reference state, i.e. t F (t ) = I ( τ = t ), L can be expressed in the updated Lagrangian description as follows: •

(2.70)

L = t F(t ) Further, t R (t )

taking

the

time-derivative

of

Eq.

(2.38)

and

noting

= t U (t ) = tV (t ) = I , it follows that •

t F (t )

•

•

•

•

= t U (t ) + t R (t ) = t V + t R (t )

(2.71)

Decomposing L into the symmetric and the skew-symmetric parts and noting Eqs. (2.69)-(2.71), it is obtained that

L = D+ W

(2.72)

2.3 Strain Rate and Spin Tensors

71

where • • T ⎫ D ≡ Ls = 1 (L + LT ) = 1 {∂v + ( ∂v ) } = t U(t ) = t V (t ) ⎪ 2 x x 2 ∂ ∂ ⎪ ⎬ v ∂v j ⎪ Dij ≡ Lijs ≡ ∂ ( j vi ) = 1 ( ∂ i + ) 2 ∂ x j ∂ xi ⎪⎭

• T ⎫ W ≡ LA = 1 (L − LT ) = 1 {∂v − ( ∂v ) } = t R (t ) ⎪ 2 2 ∂x ∂x ⎪ ⎬ vj ∂ v ∂ 1 A i ⎪ − Wij ≡ Lij ≡ ∂ [ j vi ] = ( ) 2 ∂ x j ∂ xi ⎪⎭

(2.73)

(2.74)

where D is called the strain rate tensor or the deformation rate tensor or stretching and W is called the (continuum) rotation rate tensor or (continuum) spin tensor. Here, note that D is not a time-derivative of any strain tensor but is defined independently as the rate variable although it is called the strain rate tensor. Substituting Eqs. (2.14) and (2.69) into Eqs. (2.73) and (2.74), D and W are described by U , R as follows: • • ⎫ D = 1 {F F −1 + (F F −1 )T } = 1 [(RU)• (RU) −1 + (RU) −T {(RU)• }T ] ⎪ 2 2 ⎪ • • ⎪ = 1 R ( U U −1 + U −1 U ) R T ⎪ 2 ⎬ • • W = 1 {F F −1 − (F F −1 )T } = 1 [( RU )• ( RU) −1 − (RU) −T {(RU)• }T ]⎪ 2 2 ⎪ ⎪ • • • ⎪ = R R T + 1 R{U U −1 − U −1 U}R T 2 ⎪⎭

(2.75) Consequently, we obtain ⎫ ⎪ ⎪ ⎬ • • ⎪ T 1 R T W = Ω + R ( U − U )R ⎪ 2 ⎭

• • +U T )R T D = 1 R (U 2

where

(2.76)

•

• ≡U U U −1

•

Ω R ≡ R RT

(2.77) (2.78)

72

2 Motion and Strain (Rate)

Ω R is called the relative (or polar) spin tensor. Further, D and W are described by V, R as follows:

⎫ ⎪ ⎪ • •T • T −1 • ⎪ 1 1 −1 −1 −T T = ( V V + V V ) + (V R R V − V R R V ) ⎪ 2 2 ⎬ ⎪ W = 1 [(VR )• (VR ) −1 − (VR )−T {(VR )• }T ] ⎪ 2 ⎪ • • • • = 1 (V R R T V −1 + V −1 R R T V ) + 1 (V V −1 − V −1 V ) ⎪ 2 2 ⎪⎭

D = 1 [(VR )• (VR )−1 + (VR )−T {(VR )• }T ] 2

(2.79)

and thus • • ⎫ R +Ω RT ) ⎪ +V T ) + 1 (Ω D = 1 (V 2 2 ⎪ ⎬ • •T ⎪ 1 1 T R R − W = (Ω Ω ) + ( V − V ) ⎪ 2 2 ⎭

(2.80)

where •

• ≡V V V −1 R ≡ VΩ R V −1 Ω

(2.81) (2.82)

It holds that •L

Ω L ≡ R RL T = (N (α ) ⊗ eα )• (N ( β ) ⊗ e β )T •

• (α )

= (N(α ) ⊗ e(α ) + N(α ) ⊗ e

)e( β ) ⊗ N( β )

•

= N(α ) ⊗ N(α )

(2.83)

and thus one has • (α )

N

= ΩL N (α )

(2.84)

noting e• (α ) = 0 since {e(α ) } is the fixed base. Therefore, Ω L describes the spin of the Lagrangian triad {N (α )} of the right stretch tensor U and is called the Lagrangian spin tensor. On the other hand, it holds that •

Ω E ≡ RE RE = (n (α ) ⊗ e(α ) )• (n ( β ) ⊗ e( β ) )T T

= n• (α ) ⊗ n(α )

(2.85)

2.3 Strain Rate and Spin Tensors

73

and thus one has

n• (α ) = ΩE n (α )

(2.86)

Therefore, Ω E describes the spin of the Eulerian triad {n(α )} of the right stretch tensor V and is called the Eulerian spin tensor. Here, it holds that •

R RT = (n(α ) ⊗ N(α ) )• (n( β ) ⊗ N( β ) )T • (α )

= (n• (α ) ⊗ N(α ) + n(α ) ⊗ N

)N( β ) ⊗ n( β ) • (γ )

= n• (α ) ⊗ n(α ) + n(α ) ⊗ (N(α ) • N(γ ) ) N

• (γ )

= n• (α ) ⊗ n (α ) + n (α ) ⊗ N (α ) N (γ ) ⊗ N • (γ )

= n• (α ) ⊗ n (α ) − n (α ) ⊗ N (α ) N

N ( β ) ⊗ n( β )

N( β ) ⊗ n( β )

⊗ N (γ ) N ( β ) ⊗ n ( β )

and thus the following relations hold. T T T Ω R = Ω E − R Ω L R , Ω E = Ω R + R Ω L R , Ω L = R (Ω E − Ω R ) R

(2.87) In the rigid-body rotation ( F = R, U = V = I ), it holds from Eqs. (2.73), (2.76), (2.82) and (2.87) that

L = W = Ω R = Ω E , ΩL = 0

(2.88)

In what follows consider the physical meanings of D and W . The relative velocity of the particle points P and P' , the position vectors of which are x and x + dx , respectively, is given by

dv = Ldx

(2.89)

from Eq. (2.68) and it is additively decomposed as

dv = dv r + dv d

(2.90)

where

dv d ≡ Ddx

(2.91)

dv r ≡ Wdx

(2.92)

74

2 Motion and Strain (Rate)

Here, denoting dv d in the infinitesimal line-element dx i (= dxi ei ; no sum) in the direction of the orthogonal bases ei as dv id and substituting Eq. (2.91) into D ij = D e i • e j = D d x i /dxi • e j (no sum) based on Eq. (1.88), one has

Dij =

dvid •e (no sum) dxi j

(2.93)

while dv id is not parallel to the line-element dxi in general. Therefore, Dij is the orthogonal projection of dvid /dxi (no sum) onto the e j -direction. Eventually,

Dii dxi (= dvid • ei ) (no sum) describes the relative velocity in the direction of the d line-element dx i and Dij dxi (= dv i • e j ) (i ≠ j , no sum) describes the peripheral velocity around e k (k ≠ i, j ) , as shown in Fig. 2.3. Adopting the bases (eⅠ, eⅡ, eⅢ ) which are the principal directions of D , the direction of dvId coincides with eI , i.e.

dvId ∝ e I

(2.94)

On the other hand, denoting the axial vector described in Eq. (1.121) for the skew-symmetric tensor W by w , it holds that

wi = − 1 ε rsiWrs , Wij = −ε ijr wr 2

(2.95)

and thus Eq. (2.92) is rewritten from Eq. (1.124) as

dv r = w × dx, dvir (Wis dxs = −ε isr wr dxs ) = ε irs wr dxs

d v1r

x2 dv1 d v1d

w + D 12

D12dx 1 x1

e2

dx1 P' P

D11dx1

e1

Fig. 2.3 Extension and rotation of the line-element

(2.96)

2.3 Strain Rate and Spin Tensors

75

w

D dx x

D dx x

DIdxI

xI Fig. 2.4 Deformation and rotation for principal directions of strain rate

Therefore, the arbitrary line-element dx rotates in the peripheral velocity dv r and angular velocity w , called often the spin vector, whereas 2w is called the vorticity. The total angular velocity of arbitrary line-element due to D and W is given by the equation

ω i = wi + D jk (i ≠ j ≠ k ≠ i )

(2.97)

w describes the angular velocity of line-element in the principal directions of D , i.e. the mean angular velocity since the second term in Eq. (2.97) is zero in these directions. The movement of the infinitesimal line-element dx 1 around one of the principal directions, e Ⅲ , of D is shown in Fig. 2.3. The state of deformation and rotation is shown in Fig. 2.4 for the the principal directions of D . The rate of the scalar product of the vectors dx and δ x of the infinitesimal elements connecting the three particle points P, P' , P'' with the position vectors

x, x + dx, x + δ x , respectively, is given noting Eq. (1.100) as follows:

(dx • δ x)• = d v • δ x + dx • δ v = ∂ v dx • δ x + dx • ∂ v δ x = [{∂ v + ( ∂ v)T }dx] • δ x ∂x ∂x ∂x ∂x = 2Ddx • δ x

(2.98)

76

2 Motion and Strain (Rate)

If the vicinity of the particle P undergoes the rigid-body rotation, the quantity in Eq. (2.98) for an arbitrary scalar quantity dx • δ x is zero and thus D = 0 has to hold. Inversely, if D = 0 , the quantity in Eq. (2.98) for the scalar quantity dx • δ x of arbitrary line-element vectors becomes zero and thus it can be stated that the vicinity of the particle P does not undergo a deformation. Then, D = 0 is the necessary and the sufficient conditions for the situation that a deformation is not induced, allowing only a rigid-body rotation. Denoting the lengths of the line-elements PP' and PP'' as dS and δ S and the angle contained by them as θ , it holds that

(dx • δ x )• = (dSδ S cosθ )• • ( dS )• (δ S )• = [{ + }cos θ − θ sin θ ]dSδ S dS δS

(2.99)

Further, denoting the unit vectors in the directions of the line-elements PP' and PP'' as μ and ν , respectively, and noting dx = μdS , δ x = ν δ S , it holds from Eqs. (2.98) and (2.99) that

) {(dS dS

•

+

• (δ S )• } cos θ − θ sin θ = 2Dμ • ν ( = 2 Drs μr νs ) (2.100) δS

If the particles P' and P'' mutually coincide (θ = 0) , it holds from Eq. (2.100) that (dS )• = Dμ • μ dS

(2.101)

The left-hand side of Eq. (2.101) designates the rate of extension of the line-element. Therefore, the rate of extension is given by the normal component of D in the relevant direction, noting Eq. (1.88). On the other hand, choosing the line-element PP'' to be perpendicular to the line-element PP' (θ = π /2) , it holds that •

− θ = 2 Dμ • ν

(μ • ν = 0)

(2.102)

The left-hand side of Eq. (2.102) designates the decreasing rate of the angle contained by the two line-elements mutually perpendicular instantaneously and is called the shear strain rate. • • Next, the relations of the rate E of Green strain tensor E and the rate e of the Almansi strain tensor e to the strain rate tensor D are formulated below. The material-time derivative of Eq. (2.40) is given by •

( dx • δ x)• = 2 E dx • δ X

(2.103)

2.3 Strain Rate and Spin Tensors

77

It is obtained from Eqs. (2.98) and (2.103) that •

Ddx • δ x = E dX • δ X

(2.104)

Furthermore, substituting Eq. (2.10) into Eq. (2.104) and noting Eq. (1.101), we have the relation of Green strain tensor to the strain rate tensor as follows: •

•

E = FT DF , D = F −T EF −1

(2.105)

which is obtained also from • • • E = 1 (FT F − I )• = 1 (FT F + FT F ) 2 2

• • = 1 {FT (F −T FT ) F + FT (F F −1 )F} 2

= 1 (FT LT F + FT LF) = 1 FT (LT + L)F 2 2 = FT DF

(2.106)

Next, the time-differentiation of Eq. (2.41)2 leads to •

e = − 1 {(F −T )• F −1 + F −T (F −1 )• } 2

(2.107)

where

∂ ( ∂X ) ∂ ( ∂X ) ∂ ( ∂X ) ∂ ( ∂X v) ∂F −1 ∂F −1 x ∂ x ∂ ∂x ∂t + v= v= + + − ∂X ∂v (F ) = ∂x ∂x ∂x ∂x ∂t ∂x ∂t ∂x −1 •

v = ∂ ( ∂X + ∂X v) − ∂X ∂ ∂x ∂x ∂x ∂t ∂x

(2.108)

The inside of the bracket ( ) in the last side of Eq. (2.108) is the material-time derivative of the initial configuration X and thus it is zero. Then, it holds that

(F −1 )• = −F −1L

(2.109)

Substituting Eq. (2.109) into Eq. (2.107), one has • e = 1 {(F −1L)T F −1 + F −T F −1L} 2

= LT {1 I − ( 1 I − F −T F −1 )} +{1 I − ( 1 I − F −T F −1 )}L 2 2 2 2 T 1 1 = L ( I − e ) + ( I − e )L 2 2

(2.110)

78

2 Motion and Strain (Rate)

from which one has the relation of the rate of the Almansi strain tensor to the strain rate tensor: •

e = D − LT e − eL

(2.111)

Equation (2.111) is rewritten as • 1 1 e = D − {(L + LT ) − (L − LT )}e − e{(L + LT ) + (L − LT )} 2 2

Consequently, we obtain •

e − We + eW = D − De − eD

(2.112)

where ∇

•

e ≡ e − We + eW

(2.113)

is called the Jaumann rate of Almansi strain tensor, while the Jaumann rate will be explained in 4.3. •

•

∇

In the initial state (F = I, E = e = 0) , it holds that E = e = e = D and thus all the strain rates mutually coincide. In what follows, let D and D dt be designated as ε• and dε , respectively. If the direction of the material line-element always coincides with the xⅠ -axis, the principal strain rate in this direction is given by •

∂ uⅠ • ∂ uⅠ εⅠ = ∂ uⅠ = = ∂XⅠ ∂xⅠ ∂ ( XⅠ + uⅠ) ∂u 1+ Ⅰ ∂XⅠ •

•

(2.114)

The time-integration of Eq. (2.114) leads to

εⅠ = 1n(1+

∂uⅠ ) = 1n ∂∂XxⅠⅠ = 1n(1+εⅠ) ∂XⅠ

(2.115)

Therefore, the integration of principal strain rate, εⅠ , differs from the infinitesimal (nominal) strain εⅠ . Putting ∂XⅠ → l 0 , ∂xⅠ → l , ∂uⅠ → l − l 0 , where l 0 and l are the lengths of the line-element in in the initial and the current states, respectively, it holds that

2.3 Strain Rate and Spin Tensors

79

εⅠ = 1n

l l − l0 l − l0 0 = 1n(1+ 0 ), εⅠ = l l l0

(2.116)

As described above, the material-time integration of the strain rate ε•Ⅰ leads to the logarithmic (or natural) strain. On the other hand, the infinitesimal strain coincides with the nominal strain. It results that l = 0 : εⅠ = −1 and l =2l 0 : εⅠ = 1

in

the

nominal

strain,

whereas

l = 0 : εⅠ = −∞ and

l =2l : εⅠ = 1n e 2 ( ≅ 0.693) in the logarithmic strain. Then, the nominal strain would be impertinent to use for constitutive equation describing the large deformation. In the logarithmic strain one obtains 0

ln

∫l

0

l1 dl l 2 dl l n dl dl = ∫ 0 +∫1 + " + ∫ n −1 l l l l l l l

i.e.

1n(

1 2 n ln ) = 1n( l0 ) + 1n( l1 ) + " + 1n( ln−1 ) 0 l l l l

(2.117)

Consequently, the superposition rule

εⅠ0～n = εⅠ0～1 + εⅠ0～2 + " + εⅠn 1～n −

(2.118)

holds, while εⅠa～b designates the normal strain in the xⅠ -direction when the length a

of the line-element changed from l to l b , provided that the principal direction of strain rate is fixed. When the direction of principal strain rates are fixed, it holds that

l l lⅢ 1n ( ⅠⅡ ) = 1n( lⅠ0 ) + 1n( lⅡ0 ) + 1n( lⅢ0 ) 0 0 0 lⅠlⅡlⅢ lⅠ lⅡ lⅢ

(2.119)

where lⅠ, lⅡ, lⅢ are the lengths of line-elements in the directions of three principal

strain rates and thus the logarithmic volumetric strain εν is given by the sum of the principal strains:

Ⅲ

εν = ln v = ∑ ε J V

Ⅰ

(2.120)

J=

It will be shown in 2.5 that the time-integration of the volumetric strain rate Dv ≡ trD coincides with the logarithmic volumetric strain εν in general.

80

2 Motion and Strain (Rate)

On the other hand, in the nominal strain, the above-mentioned convenient properties are not possessed and thus the error increases as the deformation becomes large. In what follows, consider the rotation of the material line-element in the plane motion, where the velocity of particles is given by

v1 = v1 ( x1 , x2 ), v2 = v2 ( x1 , x2 ), v3 = 0

(2.121)

Substituting Eq. (2.121) into Eqs. (2.73) and (2.95), one obtains ⎫⎪ ⎬ w1 = w2 = 0, w3 = −W12 ⎪⎭

D 3 j = 0 (j =1, 2, 3)

(2.122)

e3 , w3 by eⅢ , wⅢ while e 3 is the principal direction of D , the total angular velocity ω around eⅢ is given from (2.97) as Denoting

ω = wⅢ + D12

(2.123)

By choosing D and Dn , Ds for T and Tn , Ts described in 1.13, the relation of the rate of extension and the rate of rotation is shown in Fig. 2.5. It is depicted by the circle of relative velocity with the radius [{( D11 + D22 ) / 2}2 + D122 ] centering in (( D11 + D22 ) / 2, wⅢ ) in the two-dimensional plane ( Dn , ω ).

P ''

ω (= w + D12)

P (

,

)

P' (D11, D12 + w ) 11

w (D22 , D12 + w )

P

0

D

Fig. 2.5 Circle of relative velocity

D11 + D 22 2

D

Dn

2.4 Various Simple Deformations

81

The angle contained by the line-elements PP' and PP" changes as the difference of angular velocity depicted by the difference of the ordinates of the intersecting point of the relative velocity circle and the two straight parallel lines to PP' and PP" stemming from the point P as the pole. In particular, the angle contained by the line-elements in the directions of the maximum and the minimum shear strain rates, which are mutually perpendicular momentarily, changes most quickly by DⅠ − DⅢ . On the other hand, the angle contained by the line-elements for the maximum and the minimum strain rate does not change. In Fig. 2.5 it is confirmed that the mean angular velocity is given by wⅢ (= − W12 ) which is identical to the angular velocity of the line-elements in the direction of the principal strain rates.

2.4 Various Simple Deformations Let various strain (rate) and stress (rate) described in the foregoing be shown explicitly and let their relation be described for various simple deformations. These deformations are often observed in experiments for measurement of material properties. Homogeneous and isotropic deformation is assumed therein.

2.4.1 Uniaxial Loading For a cylindrical specimen with the initial length L and the initial radius R , suppose that the length and the radius changes to l and r (Khan and Huang, 1995). Choosing the XⅠ-axis to the axial direction of cylinder, it holds that

xⅠ = λⅠXⅠ, xⅡ = λⅡX Ⅱ, xⅢ = λⅡX Ⅲ

(2.124)

where

λⅠ = l , λⅡ = λ Ⅲ = r L

R

(2.125)

from which one has

⎡λⅠ 0 0 ⎤ F =U =V = ⎢⎢0 λⅡ 0 ⎥⎥ , ⎢⎣0 0 λⅡ⎥⎦ R=I

⎫ ⎡λⅠ−1 0 0 ⎤ ⎪ ⎢ ⎥ −1 −1 2 F = ⎢ 0 λⅡ 0 ⎥ , J = detF = λⅠλⅡ⎪ ⎬ ⎢ 0 0 λ −1 ⎥ Ⅱ⎦ ⎪ ⎣ ⎪ ⎭ (2.126)

82

2 Motion and Strain (Rate)

The aforementioned measures of deformation (rate) are given as

ªλ 2 0 0 º « » C = U 2 , FT F = « 0 λ 2 0 » , «0 0 λ 2 » ¬ ¼

ªλ −2 0 0 º « » b −1 =V −2 =F −T F −1= «0 λ −2 0 » «0 0 λ −2 » ¬ ¼ (2.127)

ªλ 2 − 1 0 0 º « » 1 1 2 E = (b − I ) = « 0 λ − 1 0 » , 2 2 « 0 0 λ 2 − 1»¼ ¬

ª1 − λ −2 0 º 0 « » e = 1 (I − b −1 ) = 1 « 0 1 − λ −2 0 » 2 2 « 0 0 1 − λ −2 »¼ ¬

r

λ =

λl

　0 ⎤ ⎡ln(l/L) 0 0 ⎡lnλⅠ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ 0 lnλⅡ 0 ⎥ = ⎢ 　 0 ln(r / R ) 0 ⎥ ⎢⎣ 0 0 ln( r / R ) ⎥⎦ 0 lnλⅡ⎥⎦ ⎢⎣ 0

(2.128)

(2.129)

⎡ • −1 ⎤ ⎡• ⎤ 0 ⎥ ⎢ λⅠ 0 0 ⎥ ⎡λⅠ−1 0 0 ⎤ ⎢λⅠλⅠ 0 • • −1 ⎢ ⎥ ⎢ ⎥ ⎥⎢ • −1 −1 0 λⅡ 0 ⎥ ⎢ 0 λⅡ 0 ⎥ = ⎢ 　 0 λⅡλⅡ 0 L = F F = ⎢　 ⎥ ⎥ ⎢ • ⎥ ⎢ 0 0 λ −1 ⎥ ⎢ • Ⅱ⎦ ⎢ 　 0 0 λⅡλⅡ−1 ⎥ ⎢ 0 0 λⅡ ⎥ ⎣ ⎣ ⎦ ⎣ ⎦ ⎡• −1 ⎤ 0 0 ⎥ ⎢l l • •l ⎢ ⎥ • = ⎢ 0 r r −1 0 ⎥ = D = λ r = λ ⎢ • −1 ⎥ ⎢ 0 0 rr ⎥ ⎣ ⎦ W=0

(2.130)

(2.131)

2.4 Various Simple Deformations

83

ˆ for the description of longitudinal deformation one has For the nominal strain E 0 ⎤ ⎡ λ1 − 1 0 ⎢ ⎥ ⎢ ⎥ ( l − L ) / L 0 0 ⎡ ⎤ ⎢ 0 λ 2 −1 ⎥ 0 ⎢ ⎥ ˆ= r − R) / R E = V−I 0 ( 0 = ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 0 ( r − R) / R ⎥⎦ ⎢ 0 0 λ 2 − 1⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ (2.132) ⎡• ⎤ ⎢ l /L 0 0 ⎥ • ˆ = ⎢ 0 r• /R 0 ⎥ E ⎢ ⎥ ⎢ ⎥ • ⎢ 0 0 r /R ⎥ ⎣ ⎦

(2.133)

by which the strain rate tensor is described as •

ˆ (Eˆ + Ι ) −1 D=E

(2.134)

Denoting the axial load as F , various stresses (cf. Chapter 3) are shown as follows:

⎡ F ⎤ ⎡ F 0 ⎢ π r 2 0 0 ⎥ ⎢ λⅡ2π R 2 ⎢ ⎥ ⎢ σ = ⎢ 0 0 0⎥ = ⎢ 0 0 ⎢ 0 0 0⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎢⎣ ⎡ F ⎢λ2 A 0 ⎢ Ⅱ 0 τ = Jσ = λⅠλⅡ2 ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢⎣

⎤ ⎡ 0⎥ ⎢ 2F 0 λ A ⎥ ⎢ Ⅱ 0 0⎥ = ⎢ 0 0 ⎥ ⎢ 0⎥ ⎢ 0 0 ⎥⎦ ⎢⎣

⎤ ⎡ 0 ⎥ ⎢ F λⅠ 0 A0 ⎥ ⎢ 0⎥ = ⎢ 0 0 ⎥ ⎢ 0⎥ ⎢ 0 0 ⎥⎦ ⎢⎣

⎡λⅠ−1 0 0 ⎤ ⎡ F /(λⅡ2 A0 ) 0 ⎢ ⎥⎢ 0 Π = JF −1σ = λⅠλⅡ2 ⎢ 0 λⅡ−1 0 ⎥ ⎢ 　 0 ⎢ 0 0 λ −1 ⎥ ⎢ 　 0 0 Ⅱ ⎦⎣ ⎣

⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥⎦

⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥⎦

(2.135)

(2.136)

0 ⎤ ⎡ F / A0 0 0 ⎤ ⎥ 0 ⎥ = ⎢⎢ 　 0 0 0 ⎥⎥ 0 ⎥⎦ ⎢⎣ 　 0 0 0 ⎥⎦ (2.137)

84

2 Motion and Strain (Rate)

S = JF −1σF −T

⎡ ⎢ F 0 ⎢ λⅠA0 =⎢ 0 ⎢ 0 　　 ⎢ 0 　　 0 ⎢⎣

⎤

　0 ⎥⎥

(2.138)

⎥ 　0 ⎥ 　0 ⎥⎥ ⎦

2.4.2 Simple Shear As shown in Fig. 2.6, consider the simple shear in which the shear deformation is induced parallelly to the x1 -axis.

x1 = X1 + γ X2 , x2 = X 2 , x3 = X3

(2.139)

where γ (=2D12 ) is the engineering shear strain. Denoting the shear angle by θ , it holds that •

•

γ = tanθ , γ = θ sec2 θ

(2.140)

It holds in this situation that

γ

⎡1 ⎡ ∂xi ⎤ ⎢ F=⎢ = 0 ∂X ⎥ ⎢ ⎣⎢ j ⎦⎥ ⎢0 ⎣

1 0

0⎤ 0 ⎥⎥ , 1 ⎥⎦

⎡1 F = ⎢⎢0 ⎢⎣0 −1

−γ 1 0

0⎤ 0 ⎥⎥ , 1⎥⎦

J =detF =1 (2.141)

−

θ

γ X2

X2 = x2 X

x

e2 e1 Fig. 2.6 Simple shear

X1

x1

2.4 Various Simple Deformations

85

where the inverse tensor F −1 is derived using (1.104). The components in the third line and those in the third row are zero except for unity in the third line and the third row in all tensors appearing hereinafter for the simple shear deformation. Then, for simplicity, let them be expressed by the matrix with two lines and two rows. ⎡1 γ ⎤ F=⎢ ⎥, ⎣ 0 1⎦

⎡1 − γ ⎤ F −1 = ⎢ ⎥ ⎣0 1 ⎦

(2.142)

from which it is obtained that ⎡ • ⎤ ⎡0 − γ ⎤ ⎡ •⎤ • = ⎢0 γ ⎥ L = F F −1 = ⎢ 0 γ ⎥ ⎢ ⎥ ⎢⎣ 0 0 ⎥⎦ ⎣ 0 1 ⎦ ⎢⎣ 0 0 ⎥⎦ •

⎡ ⎤ γ D = ⎢0 1 ⎥ 2 , ⎣1 0 ⎦

•

⎡ ⎤ γ W=⎢ 0 1⎥ 2 ⎣ −1 0 ⎦

(2.143)

(2.144)

Further, it holds that

γ ⎤ ⎡1 C ( = U 2 = FT F ) = ⎢ 2⎥ ⎣γ 1 + γ ⎦

，

⎡1 + γ 2 − γ ⎤ C−1 ( = U −2 = F −1FT ) = ⎢ ⎥ 1⎦ ⎣ −γ

(2.145)

ª 1 −γ º ª1+ γ 2 γ º −1 b ( = V 2 = FF T ) = « b ( = V −2 = F − T F −1 ) = « » 2» 1¼ ¬ γ ¬ −γ 1+ γ ¼ (2.146)

1 γ E = 1 (C − I ) = 1 , 2 2 γ γ2

0 −γ = 1 ( − b−1 ) = 1 2 2 −γ γ 2

(2.147)

86

2 Motion and Strain (Rate)

ⅠⅡ

Next, derive the principal stretches λα and the eigenvectors n(α ) ( α = , in the present two dimensional state) of V in Eq. (2.19). The following relation holds from Eq. (1.125).

Vn(α ) = λα n(α ) (|| n(α ) || =|| n( β ) || = 1, n(α ) • n( β ) = δαβ )

（

where the principal stretches λα of V and U acteristic equation based on Eq. (1.151).

(2.148)

）must fulfill the following char-

V11 − λα V12 = (V11 − λα )(V22 − λα ) − V12V21 V21 V22 − λα = λα2 − (V11 + V22 )λα + V11V22 − V12V21 = λα2 − (trV )λα + det V = 0

(2.149)

where, it holds from Eqs. (2.146) and (2.140) that det V = det V 2 = 1+ γ 2 −γ 2 = 1

(2.150)

and

trV 2 = 2 + γ 2 = 2 + tan 2 θ

(2.151)

Here, denoting the principal values of the second-order tensor T in the two-dimensional state as TⅠ, TⅡ with TⅢ = 0 , it holds that

trT2 = TⅠ2 + TⅡ2 = (TⅠ + TⅡ) 2 − 2TⅠTⅡ = (trT)2 − 2det T

(2.152)

in general and thus we have

Γ ≡ trV = trV 2 + 2 det V = 4 + γ 2 = 4 + tan 2θ

(2.153)

where it holds that •

•

Γ = 1 Γ −1 2 γ γ = 2

γ • γ Γ

(2.154)

Substituting Eqs. (2.150) and (2.151) into Eq. (2.149), the principal stretches λ + and λ − are given by

λ ± = 1 (Γ ± γ ) (λ + = 1 → +∞, λ − = 1 → 0 for γ = 0 → ∞ ) (2.155) 2

2.4 Various Simple Deformations

87

Ⅲ Furthermore, multiplying the identity tensor I (= ∑ n (α ) ⊗ n (α ) ) to both sides of α =Ⅰ the last equation in Eq. (2.149), it holds that

V 2 − (t rV ) V + ( det V )I = 0

，

(2.156)

，

Substituting Eqs. (2.146) (2.150) (2.153) into Eq. (2.156), one has

V=

V 2 + ( det V )I 1 = Γ trV

(

⎡1 + γ 2 γ ⎤ ⎡1 0 ⎤ ⎢ ⎥+⎢ ⎥ 1⎦ ⎣ 0 1 ⎦ ⎣ γ

)

(2.157)

resulting in

⎡ 2 + γ 2 γ ⎤ 1 ⎡ 2 + tan 2θ tanθ ⎤ (2.158) V = Vij ei ⊗ e j , ⎣⎡Vij ⎦⎤ = 1 ⎢ ⎥ ⎥= ⎢ Γ ⎣ γ 2 ⎦ Γ ⎣ tanθ 2 ⎦ for which the inverse tensor of V is given noting Eq. (1.104) as

−γ ⎤ 1 ⎡ 2 ⎡2 V −1 = 1 ⎢ = Γ ⎣ −γ 2 + γ 2 ⎥⎦ Γ ⎢⎣ − tanθ

− tanθ

⎤ ⎥ 2 + tan θ ⎦

(2.159)

2

Substituting Eqs. (2.142), (2.159) into (2.17), R is described as follows:

− γ ⎤ ⎡1 γ ⎤ 1 ⎡ 2 γ ⎤ ⎡2 = R = V −1F = 1 ⎢ Γ ⎣ −γ 2 + γ 2 ⎥⎦ ⎢⎣0 1⎥⎦ Γ ⎢⎣ −γ 2 ⎥⎦

(2.160)

Here, setting

tan θ R ( = R12 /R11) =

γ 2

= tan θ (γ = 2 tan θ R ) 2

(2.161)

R is also expressed from Eqs. (1.88), (2.21), (2.160) and (2.161) as ⎡　 cos θ R sinθ R ⎤ R = ⎡⎣(ei • n(α ) )(e j • N (α ) ) ⎤⎦ = ⎢ ⎥ R R ⎣ −sinθ cosθ ⎦

(2.162)

It holds from Eq. (2.161) that •

γ=

•

• • γ • 2 θ• R 2( 2 θ R )θ R , θ R = = 22 γ tan = + 1 2 R 2 cos θ 2{1 + (γ / 2) } Γ

(2.163)

88

2 Motion and Strain (Rate)

The relative spin in Eq. (2.78) is given from Eqs. (2.162) and (2.163) as follows:

⎡ −sinθ R cosθ R ⎤ • R ⎡cos θ R − sinθ R ⎤ ΩR = ⎢ ⎥θ ⎢ ⎥ R R R cosθ R ⎦ ⎣ −cosθ − sinθ ⎦ ⎣sinθ ⎡0 =⎢ ⎣ −1

1⎤ • R ⎡0 θ = 22 ⎢ ⎥ Γ ⎣ −1 0⎦

1⎤ • γ 0 ⎥⎦

(2.164)

Next, denoting the expression of V in the principal direction by p V , it holds from Eq. (1.73) that p

V = Q E VQ E T

，

p

Vij = (n(i ) • er )Vrs (n( s ) • e j )

(2.165)

where p

⎡λ + 0 ⎤ V=⎢ ⎥ ⎣0 λ − ⎦

⎡n (1) • e1 n (1) • e 2 ⎤ ⎡ cosθ E sinθ E ⎤ Q E = ⎡⎣n (i ) • e j ⎤⎦ = ⎢ ⎥=⎢ ⎥ ( 2) ( 2) E cosθ E ⎦ ⎣⎢n • e1 n • e 2 ⎦⎥ ⎣− sinθ

(2.166)

(2.167)

where θ E is the rotation angle of the eigenvector n (1) , n ( 2 ) of V from the bases e1 , e 2 measured in a counterclockwise direction. On the other hand, the components of RE are given from Eq. (2.25) as follows: 3

RijE = ei • RE e j = ei • ∑ n (α ) ⊗ eα • e j = ei • n ( j ) = Q Eji , R E = Q ET α =1

(2.168) The following expressions are obtained by substituting Eq. (2.165) into Eq. (2.168). p

V = R ET VR E , p

p

Vij = R riEVrs RsjE p

T

V = R E VR E , Vij = RirE Vrs R jsE

(2.169) (2.170)

Since V1 ≥ V2 always, choosing the maximum principal value λ + in the direction

，

of n(1) , it holds from Eqs. (2.158), (2.166) (2.168), (2.170) that

2.4 Various Simple Deformations

89

2 E E E E 1 ⎡ 2 + γ γ ⎤ = ⎡ cosθ − sinθ ⎤ ⎡ λ + 0 ⎤ ⎡ cosθ sinθ ⎤ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ Γ ⎢⎣ γ 2 ⎦ ⎣sinθ E cosθ E ⎦ ⎣ 0 λ − ⎦ ⎣− sinθ E cosθ E ⎦

⎡λ + cosθ E − λ − sinθ E ⎤ ⎡ cosθ E sinθ E ⎤ =⎢ ⎥⎢ ⎥ ⎢⎣λ + sinθ E λ − cosθ E ⎥⎦ ⎣− sinθ E cosθ E ⎦

⎡λ + cos 2θ E +λ − sin 2θ E (λ + − λ − )sinθ E cosθ E⎤ =⎢ ⎥ ⎢⎣(λ + − λ − )sinθ E cosθ E λ + sin 2θ E + λ − cos 2θ E ⎥⎦

(2.171)

Substituting Eq. (2.155) into Eq. (2.171), one obtains 2 1 ⎡2 + γ γ ⎤ ⎥ Γ ⎢⎣ γ 2⎦

⎡1 ⎤ 1 2 E 2 E γ sinθ E cosθ E ⎢ 2 (Γ + γ )cos θ + 2 (Γ − γ )sin θ ⎥ =⎢ ⎥ 1 1 ⎢ 2 E 2 E ⎥ E E (Γ + γ )sin θ + (Γ − γ )cos θ γ sinθ cosθ 2 2 ⎢⎣ ⎥⎦ ⎡Γ + γ cos2θ E γ sin2θ E ⎤ = 1⎢ 2 γ sin2θ E Γ − γ cos2θ E ⎥ ⎣ ⎦

(2.172)

from which it holds that

γ 1 ⎫ = γ sin2θ E → sin2θ E = 2 ⎪⎪ Γ Γ 2 ⎬ 2 γ γ ⎪ E E (V11 − V22 =) = γ cos2θ → cos 2θ = Γ Γ ⎪⎭ and thus it can be obtained that sin θ E =

1 (1 − γ 2 Γ

)=

( Γ − γ ) /2

Γ

2 cos θ E = 1 − ( Z − ) = Z + = 1 Z− Γ Γ

−γ tan θ E = Z − = Γ 2 Z+

=

⎫ Γ ( Γ − γ ) /2 Z − = = 1 ⎪ Z+ Γ Γ ⎪

⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪⎭ (2.173)

90

2 Motion and Strain (Rate)

with Ζ ± ≡ Γ (Γ ± γ ) / 2

(2.174)

where the double signs ± take in the same order. The following relations hold for Ζ±.

Ζ +Ζ − = Γ Ζ+ Γ +γ Ζ− Γ −γ = 2 , = 2 Ζ+ Ζ−

⎫ ⎪ ⎪ ⎪⎪ ⎬ 2 2 2 2 2 γ Ζ+ + Ζ− = Γ , Ζ+ − Ζ− = Γ ⎪ ⎪ 1 + 1 = 1, 1 − 1 = γ ⎪ 2 2 2 2 Γ Ζ+ Ζ− Ζ− Ζ+ ⎪⎭

(2.175)

The substitution of Eq. (2.167), (2.168), (2.173) into Eq. (2.168) yields. ⎡ 1 ⎢Ζ − − Ζ Ζ − ⎡ ⎤ + ⎢ RE = 1 ⎢ = Γ ⎣ Ζ − Ζ + ⎥⎦ ⎢ 1 ⎢Ζ ⎣ +

⎤ − 1 ⎥ Ζ+ ⎥ 1 ⎥ Ζ − ⎥⎦

(2.176)

From Eqs. (2.154) and (2.173) we have • θ E = 1 ( γ − γ• , • E = Γ (Γ + γ ) /2 1 ( γ − γ• = − 1 γ• 1 θ 2 Γ 1) cos 2θ E 2 Γ ) Γ2 Γ2

(2.177) Substituting Eqs. (2.167), (2.168), (2.177) into Eq. (2.85), it is obtained that

⎡ −sinθ E − cosθ E ⎤ • E ⎡ cosθ E sinθ E ⎤ ΩE = ⎢ ⎥θ ⎢ ⎥ E E E E ⎣ cosθ − sinθ ⎦ ⎣− sinθ cosθ ⎦ ⎡0 −1⎤ • E 2 ⎡ 0 1 ⎤ γ• 1 Ω R =⎢ θ = Γ 2 ⎢− ⎥ ⎥ =2 ⎣1 0 ⎦ ⎣ 1 0⎦

(2.178)

The following expression of U is obtained in a similar manner to that used in Eq. (2.157)

U=

U 2 + ( det U)I 1 = trU Γ

( ⎢⎣⎡1γ

γ ⎤ ⎡1 0⎤ ⎥ +⎢ ⎥ 1 + γ 2 ⎦ ⎣0 1 ⎦

)

2.4 Various Simple Deformations

91

from which it is obtained that

⎤ sinθ ⎡ 2 γ ⎤ ⎡cos θ =⎢ U= 1⎢ ⎥ 2 2 Γ ⎣γ 2+ γ ⎦ ⎣sinθ (1 + sin θ ) / cosθ ⎥⎦

(2.179)

In order to obtain the rotation RL of the eigenvector N (α ) of U , denoting the angle measured in counterclockwise direction from e 1, e 2 to N (1) , N ( 2 ) by θ L , one has U = R LV p R L

T

(2.180)

where, setting

⎡e1 • N (1) e1 • N (2) ⎤ ⎡ cosθ L − sinθ L ⎤ RL = ⎡⎣ei • N ( j ) ⎤⎦ = ⎢ ⎥=⎢ ⎥ ⎢⎣e2 • N (1) e 2 • N (2) ⎥⎦ ⎣sinθ L cosθ L ⎦

(2.181)

and substituting Eqs. (2.166), (2.179), (2.181) into Eq. (2.180), we have L sinθ L ⎤ ⎡λ + 0 ⎤ ⎡cosθ L − sinθ L ⎤ 1 ⎡ 2 γ ⎤ = ⎡ cosθ (2.182) ⎢ ⎢ ⎥ Γ ⎣γ 2 + γ 2 ⎦ ⎣− sinθ L cosθ L ⎥⎦ ⎢⎣ 0 λ − ⎥⎦ ⎢⎣sinθ L cosθ L ⎥⎦

The substitution of Eq. (2.155) into Eq. (2.182) leads to L − γ sin2θ L ⎤ 1 ⎡ 2 γ ⎤ = 1 ⎡ Γ + γ cos2θ ⎢ Γ ⎢⎣γ 2 + γ 2 ⎥⎦ 2 ⎣ − γ sin2θ L Γ − γ cos2θ L ⎥⎦

(2.183)

from which one has

⎫ ⎪⎪ ⎬ γ2 γ (U11 − U22 =) − = γ cos2θ L → cos 2θ L = − ⎪ Γ Γ ⎪⎭

sin2θ L (= sin2θ E ) = − 2

Γ

(2.184)

and

1 ⎫ =Z ⎪ Γ − ⎪ ⎪⎪ Z 2 cos θ L = 1 − ( + ) = Z − = 1 ⎬ Γ Γ Z+ ⎪ ⎪ tan θ L = Z + = 2 γ = 1 E ⎪ Z− Γ − tan θ ⎪⎭

γ sin θ L = 1 (1 + 2 Γ

)

2

=

Z+

(2.185)

92

2 Motion and Strain (Rate)

The substitution of Eq. (2.185) into Eq. (2.181) reads:

⎡ 1 ⎤ − 1 ⎥ ⎢ Z Z + − − ⎡Z − Z + ⎤ ⎥ =⎢ RL = 1 ⎢ Γ ⎣ Z + Z − ⎥⎦ ⎢ 1 1 ⎥ ⎢ Z − Z+ ⎥ ⎣ ⎦

(2.186)

Substituting Eqs. (2.176) and (2.186) into R = RE RL T based on Eq. (2.27), it holds that

⎡Ζ + − Ζ − ⎤ 1 ⎡ Z − Z + ⎤ 1 = R= 1 ⎢ Γ ⎣Ζ − Ζ + ⎥⎦ Γ ⎢⎣−Z + Z − ⎥⎦ Γ 2

⎡ 2Z + Z − Z +2 − Z −2 ⎤ ⎢ ⎥ ⎢⎣− Z+2 + Z −2 2Z + Z − ⎥⎦

⎡ 2 γ⎤ = 1⎢ Γ ⎣ −γ 2 ⎥⎦

(2.187)

which coincides with Eq. (2.160) obtained by the different approach. Substituting Eqs. (2.162), (2.167), (2.168), (2.181) into R = RE RL T , one obtains

⎡ 　cos θ R sinθ R ⎤ ⎡ cosθ E − sinθ E ⎤ ⎡ cosθ L sinθ L ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ R R E cosθ E ⎦ ⎣− sinθ L cosθ L ⎦ ⎣ −sinθ cosθ ⎦ ⎣sinθ ⎡cosθ E cosθ L + sinθ E sinθ L cosθ E sinθ L − sinθ E cosθ L ⎤ =⎢ ⎥ L L L L E E E E ⎣sinθ cosθ − cosθ sinθ sinθ sinθ + cosθ cosθ ⎦ ⎡cos(θ E − θ L ) − sin(θ E − θ L ) ⎤ =⎢ ⎥ L L E E ⎣sin(θ − θ ) cos(θ − θ ) ⎦

(2.188)

from which the following relation is obtained.

θ E = θ L −θ R

(2.189)

The rotations of n(α ) and N (α ) are shown in Fig. 2.7. Furthermore, it is derived from Eqs (2.154) and (2.185) that •

θ L = 1 ( γ + γ• • L = Γ (Γ − γ ) / 2 1 ( γ + γ• 1 • 1) , θ 2 Γ 1) = Γ 2 γ Γ2 cos 2θ L 2 Γ (2.190)

2.4 Various Simple Deformations

93

e2 n ( ) (t)

n ( ) ( ∞)

N ( ) ( ∞)

N ( ) (t) N ( ) (0) n ( ) (0)

n ( ) ( 0) ( ) (0 ) N

N ( ) (t)

θ (t)

θ R (t)

n ( ) (t)

L

θ E (t ) π π /4

N ( ) (∞)

n ( ) ( ∞)

Fig. 2.7 Rotation of Lagrangian triad N (α ) and Eulerian triad

e1

n(α )

Substituting Eqs. (2.181), (2.190) into Eq. (2.83), we obtains

⎡ −sinθ L − cosθ L ⎤ • ⎡ cosθ L sinθ L ⎤ ΩL = ⎢ ⎥θ L ⎢ ⎥ L L L L ⎣ cosθ − sinθ ⎦ ⎣− sinθ cosθ ⎦

⎡0 −1⎤ • L 2 ⎡0 −1⎤ γ• − 1 Ω R =⎢ θ =Γ2⎢ ⎥ ⎥ = 2 ⎣1 0 ⎦ ⎣1 0 ⎦

(2.191)

It can be confirmed easily that the three kinds of spins ΩR , ΩE , ΩL fulfill the relation (2.87) by Eqs. (2.160), (2.164), (2.178) and (2.191). •r

•

l Note here that a simple relation between D and λ = λ does not hold since their

principal directions does not coincide mutually. Denoting τ = σ 12 , various stress tensors are described as follows: ⎡σ 11 τ ⎤ ⎥ ⎣ τ σ 22 ⎦

σ=⎢

(2.192)

94

2 Motion and Strain (Rate)

⎡1 − γ ⎤ ⎡σ 11 τ ⎤ ⎡σ 11 − γτ τ − γσ 22 ⎤ Π = JF −1σ = ⎢ ⎥=⎢ ⎥⎢ τ σ 22 ⎥⎦ ⎣0 1 ⎦ ⎣ τ σ 22 ⎦ ⎣

(2.193)

⎡σ11 − γτ τ − γσ22 ⎤ ⎡ 1 0 ⎤ S = JF −1σF −T = ⎢ σ22 ⎥⎦ ⎢⎣ −γ 1 ⎥⎦ ⎣ τ ⎡σ 11 − γ 2σ22 − 2γτ τ − γσ22 ⎤ =⎢ ⎥ τ − γσ22 σ22 ⎦ ⎣

(2.194)

2.4.3 Combination of Tension and Distortion Consider a thin cylindrical specimen subjected to the combination of tension and distortion described by the following equation in the polar coordinate system.

r = α R, θ = Θ + ω Z , z = λ Z

(2.195)

where ( R, Θ , Z) signifies the initial configuration, and α , ω and λ denote the proportionality factors depending on the deformation, while ω is described by the relative distortion angle φ between both ends as follows:

ω ≡φ / L

(2.196)

L being the length of the specimen. Variables depending on the deformation are given as follows:

⎡ FrR FrΘ ⎢ F = ⎢ Fθ R FθΘ ⎢⎣ FzR FzΘ

⎡ ∂r ∂r ⎤ ∂r ⎢ ∂R R ∂Θ Z ⎥ ∂ FrZ ⎤ ⎢ ⎥ ⎥ r∂θ r ∂θ ⎥ Fθ Z ⎥ = ⎢ r∂θ ⎢ ∂R R ∂Θ ∂Z ⎥ ⎥ FzZ ⎥⎦ ⎢ ∂z ∂z ⎥ ⎢ ∂z ∂Z ⎥⎦ ⎣⎢ ∂R R ∂Θ

⎡ 0 ⎤ ∂α R ∂α R ∂α R ⎤ ⎡α 0 ⎢ ⎥ ⎢ R∂Θ ⎥ ∂R ∂Z ⎢ ⎥ ⎢ ⎥ r (Θ + ω Z ) r ∂ (Θ + ω Z ) r ∂ (Θ + ω Z ) ⎥ ⎢ =⎢ ∂ = 0 α ωα R ⎥ ⎢ ⎥ ⎢ R∂Θ ∂R ∂Z ⎥ ⎢ ⎥ ⎢ ⎥ λ λ λ Z Z Z ∂ ∂ ∂ ⎢ ⎥ ⎢0 0 ⎥ λ R∂Θ ∂Z ⎦ ∂R ⎣⎢ ⎦⎥ ⎣ (2.197)

2.4 Various Simple Deformations

95

from which we have ⎡1 ⎢α ⎢ F −1 = ⎢⎢0 ⎢ ⎢0 ⎢⎣

⎤ 0 ⎥ ⎥ 1 − ωR ⎥ α λ ⎥ ⎥ 1 ⎥ 0 λ ⎥⎦

0

(2.198)

and

0 0 ⎤ ⎡α ⎢ ⎥ V = ⎢0 α cosφ + αω R sin φ λsinsφ ⎥ ⎢0 λsinφ λ cosφ ⎥⎦ ⎣

(2.199)

0 0 ⎡α ⎤ ⎢ ⎥ α sinφ U = ⎢0 α cosφ ⎥ ⎢⎣0 α sinφ αω R sin φ + λ cosφ ⎥⎦

(2.200)

where

cos φ =

λ +α (α + λ ) + (αω R ) 2

2

,

sinφ =

αω R (α + λ )2 + (αω R )2

(2.201)

R being the mean radius ( R ≅ R) of the thin cylindrical specimen in the initial state. Here, R is given by Eq. (2.162). ⎡α 2 0 ⎤ 0 ⎢ ⎥ 2 C = F F = ⎢ 0 α2 ωα R ⎥ ⎢ 0 ωα 2 R λ 2 + ω 2α 2 R 2 ⎥ ⎣ ⎦ T

ª 1 0 «α 2 « 1 b −1 = F −T F −1 = «« 0 α2 « « 0 − ωR «¬ αλ

º » » ω R » − » αλ » 2 2 ω R + 1» λ2 λ 2 »¼

(2.202)

0

(2.203)

96

2 Motion and Strain (Rate)

⎡ • ⎤ ⎢ α − ω• Z ⎥ 0 ⎢ α ⎥ ⎢ ⎥ • • • −1 ⎢ • ⎥ R ω α α L = F F = ⎢ω Z α λ ⎥ ⎢ ⎥ • ⎥ ⎢ λ ⎢0 ⎥ 0 λ ⎥ ⎢ ⎣ ⎦

(2.204)

from which we have

⎡• ⎢α ⎢α ⎢ ⎢ D = ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣

0

α• α ω• α R 2λ

⎤ 0 ⎥ ⎥ ⎥ • ωα R ⎥, 2λ ⎥ ⎥ • ⎥ λ ⎥ λ ⎥ ⎦

⎡ ⎢ 0 − ω• Z ⎢ ⎢ 0 W = ⎢ω• Z ⎢ ⎢ • ⎢ 0 − ωαR 2λ ⎣⎢

⎤ 0 ⎥ ⎥ • ωα R ⎥ (2.205) ⎥ 2λ ⎥ ⎥ 0 ⎥ ⎦⎥

Stresses in various definitions are described by the following equations, designating the normal stress σ zz and the shear stress σ rθ applied to the traverse section of the cylinder by σ and τ , respectively. ⎡0 0 0 ⎤

σ = ⎢⎢0 0 τ ⎥⎥

(2.206)

⎢⎣ 0 τ σ ⎥⎦

It holds from Eqs. (2.197), (2.198) and (2.206) that

⎡1 ⎤ 0 ⎥ ⎡0 0 0 ⎤ ⎢α 0 ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥ 1 R −1 ω ⎢ ⎥ τ Π = JF σ = α 2λ ⎢0 − 0 0 α λ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ 1 ⎢0 0 ⎥⎢ ⎥⎦ τ σ 0 λ ⎥⎦ ⎣ ⎢⎣ 0 0 ⎡0 ⎤ ⎢ ⎥ 2 2 = ⎢0 − α ωτ R αλτ − α ωσ R ⎥ ⎢0 ⎥ α 2τ α 2σ ⎣ ⎦

(2.207)

2.5 Surface Element, Volume Element and Their Rates

S = ΠF −T

97

⎡1 　0 ⎢α 0 0 ⎡0 ⎤⎢ ⎢ ⎥ 1 = ⎢0 − α 2ωτ R αλτ − α 2ωσ R ⎥ ⎢0 α ⎢ ⎢0 ⎥⎢ α 2τ α 2σ ⎣ ⎦ ⎢0 − ω R λ ⎣⎢

⎡ ⎢0 ⎢ = ⎢0 ⎢ ⎢ ⎢0 ⎢⎣

0 2 2 2 − 2αωτ R + α ω σ R

λ

2 ατ − α ωσ R λ

⎤ 0⎥ ⎥ 0⎥ ⎥ ⎥ 1 ⎥ λ ⎦⎥

⎤ 0 ⎥ ⎥ 2 ατ − α ωσ R ⎥ λ ⎥ ⎥ α2 ⎥ λ ⎥⎦

(2.208)

2.5 Surface Element, Volume Element and Their Rates Presuming that the line-elements dXa , dXb , dXc change to dxa , dxb , dxc by the deformation, the following relation holds for the volume element before and after the deformation from Eqs. (1.22), (1.42), (1.43), (2.6), (2.9) and (2.11).

dv = (dx a × dxb ) • dxc = ε ijk dxia dxjb dxkc = detdx dx1a dx1b dx1c

F1R dX Ra F1R dX Rb F1R dX Rc

= dx2a dxb2 dx2c = F1R dX Ra F2 R dX Rb F2 R dX Rc

dx3a dx3b dx3c

F1R dX Rc F3 R dX Rb F3 R dX Rc

a b c F11 F12 F13 dX 1 dX 1 dX 1

= F21 F22 F23 dX 2a dX 2b dX 2c = JdV F31 F32 F33 dX 3a dX 3b dX 3c from which one has

ρ J = det F = dv = ρ0 dV

(2.209)

where ρ0 and ρ are the initial and the current densities. On the other hand, denoting the areas and the unit normal vectors of the surface elements formed by the line-elements dX a , dXb and dx a , dxb as dA N and da , n , respectively, we have

，

98

2 Motion and Strain (Rate)

dV = [dXa dXb dXc ] = (dX a × dXb ) • dXc = dXc • NdA dv

= [dx a dxb

dxc ] =

( dx a × dx b ) • dx c

=

dx c

• nda =

F dX c

• nda =

dX c

⎫⎪ ⎬ • F nda ⎪⎭ T

(2.210) noting Eq. (1.100). The following Nanson’s formula is derived from Eqs. (2.209) and (2.210).

nda = JF −T NdA, NdA = 1 FT nda J

(2.211)

da = JF −T dA, dA = 1 FT da J

(2.212)

da ≡ nda, dA ≡ NdA

(2.213)

or

where Further, noting

∂ det d x = Δ = 1 ε ε dx b dx c pq 2 pjk qbc j k ∂ x pq derived from Eqs. (1.17) and (1.23), one has

vp (dv )• = (det d x)• = ∂ det dq x (dx pq )• = ∂ det dq x dv pq = 1 ε pjk ε qbc dxjb dxkc ∂ dx qr 2 ∂xr ∂d x p ∂d x p

1 1 = ε ijk ε abc dxjb dxkc dx ra ∂vi = ε ijk ε abc dx ar dxbj dxkc Lir 2 ∂xr 2

1 1 = δ irε ijk ε abc dxra dxjb dxkc Lir = ε ijk ε abc dxia dxjb dxkc Lvv 6 6 1 = ε ijk ε abc dxia dxjb dxkc Dvv = dvDvv 6 or •

J = tr( ∂ det F FT ) = tr{(det F )F −T FT } = tr{J (F F −1 )T } = JtrLT ∂F •

•

•

from which the following relation holds for the rate of volume element, noting trTT = t rT and t rW = 0 . •

ε v = t rD =

(dv)• J• or (dv)• = dvt rD = • dV J = dv J

(2.214)

Then, the time-integration of the volumetric strain rate leads to the logarithmic volumetric strain.

2.5 Surface Element, Volume Element and Their Rates

99

ε v ≡ ∫ t rDδ t = ln dv

(2.215)

dV

Moreover, it is obtained from Eq. (2.69) and the Nanson’s formula (2.211) that •

• −T

(nda)• = ( JF −T NdA)• = ( J F −T dA + J F )NdA • −T

= {(t rD)I + F

FT }F −T JNdA •

= {(trD)I − F −T FT }nda = {(trD)I − LT}nda

(2.216)

On the other hand, noting n• • n = 0 from n • n = 1 for the unit vector n , it holds that

(da )• = n • n(da )• = n • {(nda )• − n• da} = n • (nda)•

(2.217)

Substituting Eq. (2.216) into Eq. (2.217), one obtains the rate of the current infinitesimal area as follows:

(da)• = n{(trD) I − LT }nda

(2.218)

(da)• = {(trD) I − n • Dn}da

(2.219)

which reduce to

Further, it holds from Eqs. (2.216) and (2.219) that

n• da = (nda)• − n(da)• = {(trD)I − LT}nda − n{(trD) − n • Dn}da

(2.220)

Then, the rate of the unit normal of the current surface element is given by

n• = {(n • Dn) I − LT }n

(2.221)

Chapter 3

Conservation Laws and Stress Tensors 3 Conservation Laws and Stress Tensors

Conservation laws must be fulfilled for mass, momentum, angular momentum, etc. during a deformation. These laws are described first in detail. Then, the Cauchy stress tensor is defined and further, based on it, various stresses are derived. Introducing the stress tensor, the equilibrium equations of force and moment are formulated from the conservation rules. The virtual work principle required for the analyses of boundary value problems are also described in this chapter.

3.1 Conservation Law of Mass Denoting the field of density in a material as ρ (x, t ) , the mass in a region

v is

given as m = ∫v ρ (x, t )dv . Therefore, the following conservation law of mass must

hold. • (= (∫v ρ (x, t )dv)• ) = 0 m

(3.1)

from which, noting the Reynolds’ transportation theorem of Eq. (1.277), one has the continuity equation.

ρ• + ρ divv = 0 , ρ• + ρ ∂v r = 0 ∂ xr

(3.2)

Further, setting T (x, t ) ≡ ρφ , where φ is a physical quantity per unit mass, it holds from Eqs. (1.277) and (3.2) that

(∫v ρφ dv)• = ∫v (ρ φ + ρ• φ + ρφ ∂∂vxrr ) dv = ∫v ρ φ• dv •

(3.3)

3.2 Conservation Law of Momentum The momentum possessed by a region

v in a current state is given by

∫v ρ vdv . On

the other hand, denoting the traction (or stress vector) applied to the unit surface area of the region as t, the traction applied to the surface of the region is given as K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 101–109. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

102

3 Conservation Laws and Stress Tensors

∫a t da and, denoting the body force per unit mass as b, the body force applied to the region is given by ∫ ρ bdv . The rate of momentum has to be equivalent to the sum v of the traction and the body force applied to the region. Therefore, Euler’s first law of motion (or conservation law of momentum) is given as

(∫v ρ vdv)• = ∫a t da + ∫v ρ bdv or

∫v ρ v• dv = ∫a t da + ∫v ρ bdv

(3.4)

by virtue of Eq. (3.3).

3.3 Conservation Law of Angular Momentum The angular momentum possessed by a region v in a current state is given as ∫ ρ (x × v)dv. On the other hand, denoting the moment of traction and the moment of v

body force applied to the region as ∫a (x × t ) da and ∫v ρ ( x × b)dv , respectively, Euler’s second law of motion, i.e. conservation law of angular momentum is described as

(∫v ρ x × vdv)• = ∫a x × tda + ∫v ρ x × bdv

( ∫v ρε ijk xj vk dv)• = ∫a ε ijk xj t k da + ∫v ρε ijk xj bk dv which reduces to •

∫v ρ x × v dv = ∫a x × tda + ∫v ρ x × bdv •

∫v ρε ijk xj vk dv = ∫a ε ijk xj t k da + ∫v ρε ijk xj bk dv •

(3.5)

•

noting (x × v)• = v × v + x × v = x × v and Eq. (3.3).

3.4 Stress Tensor When the infinitesimal force vector df applies to the surface with infinitesimal area da and the unit normal vector n, the stress vector is given as shown below.

t ≡ df da

(3.6)

3.4 Stress Tensor

103

Now, introduce the following second-order tensor

σ fulfilling the relation

t = σT n, ti = σ ji n j by the quotient law described in 1.3.2. The components of the tensor by Eq. (1.88) as

(3.7)

σ are given

σij = ei • σej = σei • ej = σ Tei • ej

(3.8)

if σ is the symmetric tensor. Here, when we choose ei to the unit normal vector n of the surface on which t applies, the following equation holds by substituting Eq. (3.7) with n = ei into Eq. (3.8). (3.9) σij = t • ej Therefore, σij is the component in the direction of e j for the stress vector t applying on the surface element having the outward-normal ei . The tensor σ is called the Cauchy stress tensor. Equation (3.7) is called the Cauchy’s fundamental theorem or Cauchy’s stress principle. It holds from the conservation law of angular momentum described in 3.6 that (3.10) σ = σT (σij = σ ji ) which means that σ is the symmetric tensor. Various stress tensors are defined in addition to the Cauchy stress tensor described above. Some of them, which are often used in continuum mechanics, are presented below. The tensor τ defined by the following equation is called the Kirchhoff stress tensor. (3.11) τ = Jσ The vector t defined by the following equation is called the nominal stress vector (see Fig. 3.1).

df t ≡ dA

(3.12)

Tensor Π , which is related to t by the following equation, is called the nominal stress tensor or the first Piola-Kirchhoff stress tensor.

t ≡ ΠN (ti ≡ ΠiA NA )

(3.13)

Here, substituting Eqs. (2.211) and (3.13) into Eq. (3.12), we have

df ≡ ΠNdA = 1 ΠFΤ nda (dfi = Π iΑ NA dA = 1 Π iΑ FrA nr da ) J J

(3.14)

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3 Conservation Laws and Stress Tensors

On the other hand, the substitution of Eq. (3.7) into Eq. (3.6) yields

df = σT nda

(3.15)

Comparing Eqs. (3.14) and (3.15), it holds that

σT nda = ΠNdA

(3.16)

and the relation of σ and Π is obtained noting Eq. (2.211) as shown below.

⎫ σ = 1 FΠT (= 1 ΠFT ) (σ ij = 1 FiAΠ j Α = 1 ΠiΑ F jA ) ⎪ J

J

Π = JσF

−T

J

J

(ΠiΑ = J σ ir (F −1 )rA ) (Π ≠ ΠT )

⎬ ⎪ ⎭

(3.17)

df

df

n

N F

da da

dA dA dX

dv

dV dV

dx = FdX

Fig. 3.1 Variation of configuration

Further, the tensor S defined by the following equation is called the second Piola-Kirchhoff stress tensor. t = SN

(3.18)

where −1 t ≡ F df = F −1t dA

(3.19)

Using Eq. (2.211) into these equations, one has the following expression.

df ≡ FSNdA = 1 FSFT nda J

(3.20)

3.5 Equilibrium Equation

105

Comparing Eqs. (3.15) and (3.20), it holds that

⎫ ⎪ ⎬ = J (F −1 ) Ai σ ij (F −1 )jB ) (S = S T )⎪ ⎭

σ = 1 FSF T (σ ij = 1 FiA S AB FBj ) J

J

S = J F σF −1

−T

( S AB

(3.21)

The following stress is called the convected stress. cσ ≡ FT σF ( cσ = cσT )

(3.22)

The relations of various stress tensors defined above are summarized in Table 3.1. Table 3.1 Relations of various stress tensors Names, Notations Cauchy

ı (= ıT )

Ĳ (= ĲT )

Ĳ Nominal (1st Piola-Kirchhoff) Ȇ 2nd Piola-Kirchhoff S Convected cı

S (= ST )

c ı ( = cı T )

1 FȆT J

1 FSFT J

F −T cı F −1

FȆT

FSFT

JF −T cıF −1

FS

JF −T cıF −1F −T

1 Ĳ J

ı Kirchhoff

Ȇ ( ≠ ȆT )

Jı

JıF −T

ĲF −T

JF −1ıF −T

F −1ĲF −T

Ȇ T F −T

F T ıF

1 F T ĲF J

1 F T FȆT F J

JF −1F −T cıF −1F −T 1 F T FSFT F J

−1 (Note) t ≡ df , t ≡ df , t ≡ F df = F −1t da dA dA t = σn , Jt = τn , t = ΠN , t = SN

3.5 Equilibrium Equation Substituting Eq. (3.7) into Eq. (3.4) for the conservation law of momentum and noting Eq. (3.10), the following equation is obtained.

(∫v ρ vdv)• = ∫a σT nda + ∫v ρbdv, (∫v ρ vi dv)• = ∫a σi r n r da + ∫v ρbidv (3.23) The right had side is given from Eq. (3.3) as

(∫v ρ vdv)• = ∫v ρ v• dv

106

3 Conservation Laws and Stress Tensors

and the first term in the right-hand side of Eq. (3.23) is given by Eq. (1.267) of Gauss’ divergence theorem as

σ

∫a σi r nr da = ∫v ∂∂xirr dv By this equations and Eq. (3.3) the local form of Eq. (3.23) is given as

σ • ∇ + ρ b = ρ v• ,

∂σ ij + ρ bi = ρ v• i ∂x j

(3.24)

This equation is called the Cauchy’s first law of motion, i.e. the equilibrium equation. On the other hand, substituting Eqs. (2.209) and (3.16) into Eq. (3.23), one has

(∫V ρ0 vdV )• = ∫A ΠNd A + ∫V ρ0bdV

(3.25)

which is rewritten by Gauss’ divergence theorem as follows:

∫V ρ

0

v• d V = ∫V Π • ∇X dV + ∫V ρ 0bdV

(3.26)

where ∇ X ≡ (∂ / ∂X A )e A = ∂/∂ X . The local form of this equation is given as • • Ȇ • ∇X+ ρ0b = ρ 0 v, ∂Ȇ iA + ρ0bi = ρ 0 vi ∂XA

(3.27)

The equilibrium equation in a rate form is required in constitutive equations for irreversible deformation including elastoplastic deformation. The time-differentiation of Eq. (3.27) engenders the following rate-type (or incremental-type) equilibrium equation, provided that the acceleration does not change, i.e. •• v = 0. •

•

Ȇ • ∇X + ρ0 b = 0, Denoting

•

∂ ȆiA + ρ • = 0 0 bi ∂XA

∗) )∗ 1 • 1 • ı ≡ J Ȇ FT (≠ ı T = J F ȆT )

(3.28)

(3.29)

and noting Eq. (3.17), we have • •− •− • ∗ − σ = 1J ( J•σF T + J σ• F −T + J σ F T) FT = JJ σ + σ+ σ F T FT

Substituting (F −T F T )• = F• −T F T + F −T F• T = F• −T F T + LT = 0 due to Eq. (2.69) and Eqs. (2.73) and (2.214) to this equation, the following holds.

3.6 Equilibrium Equation of Moment

107

: ∗ • σ = σ + σ t r L − σ LT = σ + σ t r D − σ D + W σ

(3.30)

∗ : • Therein, σ is designated as the nominal stress rate, whereas σ ≡ σ − w σ + σ w is the Jaumann stress rate which will be described in detail in the next chapter. The partial derivative of Eq. (3.29) by x j noting ∂ ( F jA / J ) / ∂ x j = 0 (see Appendix 2) leads to the following.

* • • • ∂ σ ij = 1 F ∂ΠiΑ = 1 ∂ΠiΑ ∂x j = 1 ∂ΠiΑ jA J J ∂x j ∂X A J ∂X A ∂x j ∂x j Substitution of this relation into Eq. (3.28) yields the rate-type equilibrium defined in the current configuration: * • • ∗ ∂ σ ij ı • ∇ + ρ b = 0, ∂x + ρ bi = 0 (3.31) j

This equation is derived also by the following manner. From Eq. (3.23) one has

(∫v ρ v• dv)• = (∫a σT nda)• + (∫v ρbdv)• (∫V ρ 0 v• dV )• = (∫a σT nda)• + (∫V ρ 0 bdV )• •

•

0 = ³a ı T nda + ³a ıT (nda )• + ³ ρ0 b dV V

•T

0 = ³a ı nda + ³a ı

T {(trD) I

T } nda +

−L

•

³v ρ b dv •

• T 0 = ³v (ı T + ıT trD − ıT L ) • ∇dv + ³v ρ b d v

which results in Eq. (3.31), noting Eqs. (1.274), (2.216) and σ = σT .

3.6 Equilibrium Equation of Moment Substituting Eq. (3.7) into Eq. (3.5) of the conservation law of angular momentum and noting (3.10), one has •

∫v ρε ijk xj v k dv = ∫a ε ijk xj σ kr nr da + ∫v ρε ijk xj bk dv

(3.32)

Because the first term in the right-hand side of this equation is rewritten as

∫a ε ijk xj σ kr nr da = ∫a ε ijk

∂ xj σ kr σ dv = ∫v (ε ijkσ k j + ε ijk xj ∂ xkr ) dv ∂x r ∂ r

108

3 Conservation Laws and Stress Tensors

Eq. (3.32) leads to

∫v{ε ijkσ k j + ε ijk xj (∂∂σxkrr + ρbk − ρ v• k )}dv = 0 Noting the equilibrium equation (3.24) to this equation, it holds that ε ijkσ k j = 0 from which we have the symmetry of Cauchy stress tensor, i.e.

σ ij = σ ji

(3.33)

3.7 Virtual Work Principle The stress (rate) field fulfilling the equilibrium equation and the boundary condition of stress is called the statically admissible filed. On the other hand, the displacement (velocity) field fulfilling the geometrical requirement FiA (= ∂xi / ∂X A ) = δ iA + ∂ui / ∂X A or Dij = (∂vi / ∂x j +∂v j / ∂xi ) / 2 and the boundary condition of displacement (velocity) is called the kinematically- admissible field. Denoting arbitrary statically admissible stress field and kinematically-admissible velocity field by ( )Δ and ( ) ∇ respectively, one has the following equation from Eq. (3.24).

³v (ı Δ• ∇

+ ρ b − ρ v• ) • u∇ dv = 0,

Δ

∂σ ij ³v ( ∂xj

+ ρ bi − ρ v• i ) u∇i dv = 0 (3.34)

Using the Eq. (1.264) of Gauss’ divergence theorem, we have

∫v σ ijΔ

Δ

Δ

∂σ ij ∇ ∂ (σ ij u ∇i ) ∂ u ∇i u i dv dv = ∫ dv − ∫ v v x x ∂ j ∂ j ∂x j

∂σ ijΔ ∇ u i dv v ∂x j

= ∫a σ ij u∇i nj da + ∫ σ ijΔ ui nj da − ∫ av

t

∂σ ijΔ ∇ u i dv v ∂x j

= ∫a tij u∇i da + ∫ σ ijΔ nju i da − ∫ t

av

where ( − ) designates the given boundary condition, and at and a v specify the surfaces of the body on which the traction (rate) and the displacement (velocity) are given, respectively. Substituting Eq. (3.34) into this equation, the following virtual work principle described by the quantities in the current state is obtained.

∂ u∇

∫v σ ijΔ ∂xji dv = ∫at ti u∇i da + ∫av σ ij nj ui da + ∫v ρbiu∇i dv − ∫v ρ vi ui dv Δ

•

∇

(3.35)

3.7 Virtual Work Principle

109

Similarly, the following virtual work principle can be described by the quantities in the initial state from Eq. (3.27).

∂ u∇

∫V Π iAΔ ∂XiA dV = ∫At ti u∇i dA + ∫Av Π iA NJ ui dA + ∫V ρ biu∇i dV − ∫V ρ Δ

0

0

v• i u∇i dV (3.36)

Furthermore, one has the rate-type virtual work principle from Eqs. (3.28) and (3.31) as follows:

• Δ

•

∫vσ ij Dij dv = ∫at t i v∇i da + ∫av σ ij nj vi da + ∫v ρ bi v∇i dv ∇

•

•Δ

•Δ •∇

•

•Δ

∫V ΠiA F iA dV = ∫At t i vi∇ d A+ ∫Av Π iA NA vi dA + ∫V ρ

•

0

b i vi∇ dV

(3.37)

(3.38)

Chapter 4

Objectivity and Corotational Rate Tensor 4 Objectivity and Corotatio nal Rate Tensor

The mechanical properties of a material are independent of the reference frame of the observer. Therefore, a constitutive equation describing the properties must be invariant under the change of coordinate system mutually rotating with respect to each other. The physical meaning of this type of the objectivity of constitutive equations and the general form of corotational rate of a tensor that has to be used in stead of the usual material-time derivative (under the fixed coordinate system) in order to fulfill the invariance are described in this chapter.

4.1 Objectivity When a material is subjected to a constant stress, i.e. a stress the components of which are observed to be constant from the material itself, the deformation has not to be induced even if a rigid-body rotation is added. However, the components of stress in a coordinate system fixed in the laboratory change so that the stress rate is observed to be induced in the fixed coordinate system if the material is subjected to a rotation. On the other hand, the strain rate described in 2.3 is given by the symmetric part of the velocity gradient excluded its skew-symmetric, i.e. rotational part and thus its components are not influenced by the rotation. The one-to-one correspondence between the stress and the strain does not exist, and thus the stress rate and the strain rate have to be related mutually in an irreversible constitutive equation leading to the rate-type equation, e.g. the viscoplastic, the elastoplastic and the viscoplastic constitutive equations. Therefore, if the material-time derivative of stress observed in the fixed coordinate system is adopted in these constitutive equations, the incorrect prediction is given such that the strain rate is induced and thus the material deforms even if the stress observed by material itself is constant. Let the general interpretation for this fact and how to remedy this defect be considered below. As already mentioned, the nechanical properties of materials are independent of the mutual rigid body motion of observers. Therefore, the deformation characteristics must be described uniquely in an identical equation independent of the relative position and motion to the coordinate systems by which it is described. This fact is called the principle of objectivity or principle of material-frame indifference or simply objectivity (Oldroyd, 1950). The explicit problem occurs in the deformation analysis of materials subjected to the rotation, while the coordinate system describing a constitutive equation is fixed usually. Therein, the check is K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 111–125. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

112

4 Objectivity and Corotational Rate Tensor

required as to whether or not the deformation behavior is formulated such that it is not influenced by the rotation. As the preliminary examination, how the physical quantities relevant to the stress and deformation are influenced by the rotation will be first considered in the next section. Consequently, how to remedy this defect for the quantity, which is observed to be constant by the material itself but is observed to change in the fixed coordinate system, will be considered. Further, the explicit mathematical process that the material-time derivative of arbitrary scalar-valued tensor function is transformed directly to its corotational derivative is shown, while this transformation is required in the consistency condition which is derived from the yield function on the formulation of plastic constitutive equation described in the later chapter. Mechanical elements are often subjected to a rotation independent of the occurrence of deformation, as seen in gears, wheels, etc. in the case of machinery. Soils near the side edges of footings, at the pointed ends of piles, etc. undergo a large rigid-body rotation in soil structures. Therefore, a formulation of constitutive equations independent of rigid-body rotation is of great importance in practical engineering problems.

4.2

Influence of Rigid-Body Rotation on Various Mechanical Quantities

In order to check whether or not a constitutive equation is formulated in accordance with the objectivity principle, it is expedient first to examine the influence of rigid-body rotation on variables used in constitutive equations. To this end, it is sufficient to derive the transformation rules for the components of constitutive variables under the change of coordinate system from one with the fixed base {ei } into the other with the rotating base {e∗i (t ) = QT (t )ei } , which coincide with base {ei } in initial instant and then rotates with the elapse of time. Designating the infinitesimal line-element vectors observed in the above-mentioned coordinate systems before and after the deformation as dX, dx and dX∗ , dx∗ , the relations

dX∗ = dX (Q = I for t = 0), dx∗ = Qdx

(4.1)

hold where Q = ei ⊗ e∗i as shown in Eq. (1.77) and they are related by

dx = FdX, dx∗ ( = Qdx = QFdX) = F∗dX

(4.2)

and consequently, we have F* = QF

(4.3)

It is known from Eq. (4.3) that the deformation gradient F transforms as the second-order tensor between two fixed coordinate systems, but it would be observed as if it were a vector when a rigid-body rotation is superposed. In other

4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities

113

words, it obeys the objective transformation as the first-order tensor for the superposition of rigid-body rotation. It is called the two-point tensor because it is described using the two bases in the initial and the current states. Substituting Eq. (4.3) into Eqs. (2.15)-(2.16) and (2.41), the following relations hold for various quantities describing a deformation. R* = QR

(4.4)

U∗ = U, C∗ = C (F∗T F∗ = (QF)T QF = FT QT QF = FT F)

(4.5)

T T T T V∗ = QVQT , b∗ = Q bQT (F∗F∗ = QF(QF) = QFF Q ) (4.6)

E∗ = E , e ∗ = QeQT

(4.7)

Noting the relation •

•

•

•

(QF)• (QF)−1 = (QF + QF)F −1Q −1 = Q(FF −1 − QT Q)QT •

and Eq. (4.3), it holds for the velocity gradient L (≡ FF −1) that

L∗ = Q(L − Ω)QT = QLQT + Ω

(4.8)

where Ω and Ω are given by • • ⎫ Ω ≡ QT Q, Ω ≡ Q ri Qrj e i ⊗ e j ⎪ ⎪ ⎬ • •T Ω ≡ Q Q , Ω ≡ Qir Q jr e i ⊗ e j ⎪ ⎪⎭

(4.9)

and they are related by •

Ω ≡ Q QT = − QΩQT

(4.10)

•

where Q is given by • Q = er ⊗ e•∗r

(4.11)

Substituting Eq. (4.11) into Eq. (4.9) 1 , we have

Ω ≡ e•*r ⊗ e*r

(4.12)

e• *i = Ω e*i

(4.13)

from which it is obtained that It is seen from Eq. (4.13) that Ω is the spin of the base {e∗i } .

114

4 Objectivity and Corotational Rate Tensor

The substitution of Eq. (4.8) into Eqs. (2.73) and (2.74) yields the following relations. D∗ = QDQT

W∗ = Q (W − Ω)Q T = QWQ T + Ω

(4.14) (4.15)

The conclusions concerning the influence of rigid-body rotation and following from Eqs. (4.5)-(4.15) are: 1) The right Cauchy-Green deformation tensor C and the Green strain tensor E are observed to be unchangeable, obeying the transformation rule of scalar quantities independent of the rigid-body rotation. On the other hand, the left Cauchy-Green deformation tensor b and the Almansi strain tensor e obey the transformation rule of second-order tensors. 2) The strain rate tensor D obeys the transformation rule of the second-order tensor. On the other hand, the velocity gradient tensor L and the continuum spin tensor W are directly subjected to the influence of rate of rigid-body rotation. The following relations hold for stress tensors described in Chapter 3.

σ ∗ = QσQT (t ∗ = Qt = Qσn = QσQT n ∗ = σ ∗n ∗) τ ∗ = QτQT

(4.16) (4.17)

Π ∗ = QΠ

(Π∗ = Jσ∗F∗−T = JQσQT (QF) −T = JQσQT Q −T F −T = QJσF −T = QΠ) (4.18)

S∗ = S (S ∗ = F ∗ 1Π ∗ = (QF ) −1 (QΠ ) = F −1QT QΠ = F −1Π = S ) −

(4.19) Then, the Cauchy stress tensor σ and the Kirchhoff stress tensor τ are observed as the second-order tensor. On the other hand, the nominal stress tensor Π is observed as the first-order tensor which is the two-point tensor and the second Piola-Kirchhoff stress tensor S is observed as the scalar under the superposition of rigid-body rotation.

4.3 Rate of State Variable and Corotational Rate Tensor It was stated in 1.15 that the material-time derivatives of state variables pursuing the material particle must be used in rate-type constitutive equations. In addition, as described in 4.1 for the stress rate tensor, if the material-time derivative of physical

4.3 Rate of State Variable and Corotational Rate Tensor

115

quantity observed in the fixed coordinate system is adopted in a constitutive equation, the irrational prediction is resulted. Instead of the material-derivative, considering the rate of the quantity which is observed not only moving but also rotating with the material, we would have to adopt its translation to the fixed coordinate system describing the constitutive equation. The explicit formulation for this quantity will be shown below (Hashiguchi, 2007). Consider the transformation of the material-time derivative of a state variable obeying the objective transformation (1.59) or (1.61). Their material-time derivative reads: •

• T ∗p1 p2 ⋅⋅⋅ pm = Q p1q1 Q p2 q2

+ Q p1q1 Q p2q2 •

• • •

•

Q pm qm Tq1q2 ⋅⋅⋅qm + Q p1q1 Q p2q2 • • • Q pm qm Tq1q2 ⋅⋅⋅qm + • • •

• • •

•

Q pm qm Tq1q2 ⋅⋅⋅qm + Q p1q1 Q p2q2

• • •

•

Q pm qm T q1q2 ⋅⋅⋅qm (4.20)

•

•

T p1 p2 ••• pm = Q q p Q q p • • • Q qm pm Tq q ••• q + Qq p Q q p • • • Qq m pm T ∗q1q ••• qm + • • • m 2 1 1 2 2 1 2 1 1 2 2

+ Qq1 p1 Qq

•

p • • • Q qm pm

2 2

T ∗q1q2 ••• qm + Qq1 p1 Qq2 p2

•

• • • Qq p T ∗ m m q1q2 ••• qm

(4.21) •

•

•

• Noting the relation Q pi qi = δ pi s Q s qi = Q pit Qst Q s qi = − Q pit Q st Qsqi = − Q pit Ωt qi and replacing t → qi , qi → ri , then Eqs. (4.20) and (4.21) can be rewritten as

follows: •

T ∗p1 p2 ⋅⋅⋅ pm = Q p q Q p q 1 1 2 2

•

T p1 p2 ••• pm = Qq p Qq p 1 1 2 2

•

Q pm qm (T q1q2 ⋅⋅⋅qm − Ωq r Tr q ⋅⋅⋅q − Ωq r Tq r ⋅⋅⋅q 2 2 11 1 2 1 2 m m (4.22) − • • • − Ωq rmTq q ⋅⋅⋅r ) 1 1 2 m

• • •

• • • • Qq p (Tq∗q ••• q 1 2 m m m

− Ω q1 r1Tr∗q2 ••• q − Ωq r Tq∗r ••• qm 1 2 2 1 2 m

− • • • − Ω qm rm Tq∗1q2⋅⋅⋅rm )

(4.23)

It is known from Eqs. (4.20)-(4.23) that the material-time derivative cannot be •

adopted in constitutive equations, since the components T p1 p2 ••• pm in the fixed •

coordinate system changes even when the components T ∗p1 p2 ⋅⋅⋅ pm observed in the coordinate system rotating with the material does not change. Eqs. (4.20)-(4.23) are expressed for scalar S , vector v and second-order tensor T in symbolic notation as follows: •

•

S∗ = S v• ∗ = Q ( v• − Ω v) , v• = QT ( v• ∗ − Ω v ∗)

(4.24) (4.25)

116

4 Objectivity and Corotational Rate Tensor •

•

•

•

T ∗ = Q ( T − Ω T + TΩ) Q T , T = Q T (T ∗ − ΩT∗ − T ∗ Ω )Q

(4.26)

,

Now, consider the tensor T having the components obtained using the inverse transformation from the components observed by the coordinate system rotating with the material. Then, using Eq. (4.22), it is represented by ,

T p1 p2 ••• pm = Qq1 p1 Qq2 p2 = Qq1 p1 Qq2 p2

• • • Qq p m m

•

T ∗q1q2 ••• qm

• • • Qq p Q Q q1t1 q2t2 m m

• • •

Qqm t m

•

(T t1t2 ⋅⋅⋅tm − Ωt r Tr t ⋅⋅⋅t − Ω t2 r Tt r ⋅⋅⋅q − • • • − Ω t rmTt t ⋅⋅⋅r ) m m m 11 1 2 1 1 2 2 1 2 = δ p1t1δ p2 t 2 • • • δ pm t m •

(T t1t2 ⋅⋅⋅tm − Ωt r Tr t ⋅⋅⋅t − Ωt2 r Tt r ⋅⋅⋅t m − • • • − Ω t rmTt t ⋅⋅⋅r ) 11 1 2 m 2 1 2 1 12 m •

= T p1 p2 ••• pm − Ω p1r1 Tr1 p2 ••• pm − Ω p2r2T p1r2 ••• pm − • • • − Ω pmrmTp p2 •••rm 1 (4.27) While various spins describing explicit rotational rates of material would be assumed, denoting them by the symbol ω , let the following corotational rate tensor , D T be introduced, following the form of T in Eq. (4.27), i.e.

D • T p1 p2⋅⋅⋅pm = T p1 p2⋅⋅⋅pm − ω p1r1 Tr1 p2⋅⋅⋅pm − ω p2 r2T p1r2⋅⋅⋅pm − • • • − ω pmrm Tp1 p2⋅⋅⋅rm

(4.28)

which must obey the objective transformation rule

D T *p1 p2⋅⋅⋅ pm = Q p1q1 Q p2 q2

• • •

D Q pm qm T q1q2⋅⋅⋅qm

(4.29)

between an arbitrary and the reference coordinate systems with the bases {e∗i } and {ei } . Thus, substituting Eq. (4.28) into Eq. (4.29) and noting Eq. (4.22), it must hold that •

T *p1 p2⋅⋅⋅pm − ω *p1r1 Tr1*p2⋅⋅⋅pm − ω *p2 r2T p*1r2⋅⋅⋅pm − • • • − ω*pmrm Tp*1 p2⋅⋅⋅rm = Q p1q1 Q p2q2

• • •

Q pm qm

•

(T q1q2⋅⋅⋅qm − Ωq r Tr q ⋅⋅⋅q − Ωq2r Tq r ⋅⋅⋅q − • • • − Ωq rmTq q ⋅⋅⋅r ) 11 1 2 1 2 1 1 2 m m m 2 −ω *p1r1 Qr1q1Q p2q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm − ω *p2 r2 Q p1q1Qr2q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm

4.3 Rate of State Variable and Corotational Rate Tensor

− • • • − ω*pm rmQr1q1Q p2q2 = Q p1q1 Q p2q2

• • •

• • •

117

Q pm qm Tq1*q2⋅⋅⋅q m

Q pm qm

•

(T p1 p2⋅⋅⋅pm − ωq1r1 Tr1q2⋅⋅⋅qm − ω q2 r2Tq1r2⋅⋅⋅qm − • • • −ωqmrm Tq1q2⋅⋅⋅rm) from which it obtained that

Q p1q1 Qp2q2

• • •

Q pm qm (Ωq r Tr q ⋅⋅⋅q + Ωq r Tq r ⋅⋅⋅q + • • • + Ωqm rmTq q ⋅⋅⋅r ) 2 2 11 1 2 1 2 1 2 m m m

+ ω *p1r1 Qr1q1Q p2q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm +ω *p2 r2 Qp1q1Q p2r2

+ • • • + ω*pmrmQ p1q1Q p2 q2 = Q p1q1 Q p2 q2

• • •

• • •

• • •

Q pm qm Tq1r2⋅⋅⋅qm

Q pm rm Tq1*q2⋅⋅⋅qm

Q pm qm

( ωq1r1 Tr1q2⋅⋅⋅qm + ω q2 r2Tq1r2⋅⋅⋅qm + • • • + ωqm rm Tq1q2⋅⋅⋅rm ) which may be arranged into the form

ω *p1r1 Qr1q1Q p2q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm

= Q p1q1 Q p2q2

• • •

= Q p1α Q p2 q2

• • •

Q pm qm ( ωq1r1 − Ωq r )Tr1q2⋅⋅⋅qm 11

Q pm qm (ωα q1 − Ωα q1)Tq1q2⋅⋅⋅qm

= Q p1α δq1β Q p2 q2

• • •

Q pm qm (ωαβ − Ωαβ )Tq1q2⋅⋅⋅qm

= Q p1α Qr1 β Qr1q1Q p2 q2

• • •

Q pm qm ( ωαβ − Ωαβ )T q1q2⋅⋅⋅qm

= Q p1α Qr1 β ( ωαβ − Ωαβ )Qr1q1Q p2 q2

• • •

Q pm qm T q1q2⋅⋅⋅qm

Eventually, ω must obey the following transformation rule.

ω∗ = Q(ω − Ω)QT , ωij∗ = Qir Q js (ωrs − Ωrs ) The following equations for scalar S , vector hold from Eqs. (4.28).

v and the second-order tensor T

D • D S = S = S∗

(4.31)

vD = v• − ω v ; vD ∗ = Q vD ; vD = Q T vD ∗ D

•

D

D

(4.30)

D

(4.32)

D

T = T − ω T + Tω ; T ∗ = Q T QT , T = QT T ∗ Q

(4.33)

118

4 Objectivity and Corotational Rate Tensor

Now, introduce the symbols for the coordinate transformation.

(Q aT b) p1 p ⋅⋅⋅ pm ≡ Q p q Q p q • • • Q p q Tq q q ⎫ 2 m m 1 1 1 2 ⋅⋅⋅ m ⎪ 2 2 ⎬ (QT a Tb) p1 p2⋅⋅⋅ pm ≡ Qq p Qq p • • • Qq p Tq1q ⋅⋅⋅qm ⎪ 2 m m ⎭ 1 1 2 2

(4.34)

Then, the following expressions for an arbitrary tensor T hold from Eqs. (1.59) and (4.29). T = Q T a T ∗b

T∗ = Q a T b,

(4.35)

and D

D

D

T∗ = Q a T b ,

D

T = QT a T∗b

D

•

D

•

( T∗ = T∗, T = QT a T∗ b ) (4.36)

The tensor is calculated by the following time-integration by Eq. (4.28), (4.32) and (4.33).

Tp1 p2⋅⋅⋅ pm = (TD p p p + ω p r T ∫ 1 2⋅⋅⋅ m 1 1 r1 p2⋅⋅⋅pm + ω p2 r2T p1r2⋅⋅⋅pm

+ • • • + ω pm rm Tp1 p2⋅⋅⋅rm ) dt

(4.37)

i.e.

v = ∫ ( vD + ω v )dt D T = ∫( T + ωT − Tω ) dt

(4.38) (4.39)

D

after calculating T p1 p2⋅⋅⋅pm or vD or T by the constitutive relation. D , While the corotaional rate T in Eq. (4.28) following the tensor T in Eq. (4.27) is introduced above, note that various objective corotational rate have D , as will be described in 4.5 for stress been proposed, which do not belong to T rate. When the continuum spin W in Eq. (2.74) is adopted for the material spin ω, D is called the Jaumann rate (Jaumann, 1911) and then the corotational rate tensor T : let it be designated by the symbol T , i.e. D

:

•

T p1 p2⋅⋅⋅ pm = T p1 p2⋅⋅⋅ pm − Wp1r1 Tr1 p2 ••• pm − Wp2r2 T p1r ⋅⋅⋅ pm − • • • 2 − W pm rm Tp1 p ⋅⋅⋅r 2

m

which is represented for the second-order tensor as

(4.40)

4.4 Transformation of Material-Time Derivative

119

•

:

T = T − WT + T W

(4.41)

The spin W obeys the transformation rule (4.30) as shown in Eq. (4.15). Furthermore, the corotational rate tensor adopting the relative spin Ω R ( = R• RT ) in Eq. (2.78) is called the Green-Naghdi rate tensor (Green-Naghdi, 1965) or ⊕ Dienes rate tensor (Dienes, 1979) and is represented by T , i.e. •

⊕

T p1 p2⋅⋅⋅pm = T p1 p2⋅⋅⋅pm − ΩpR1r1 Tr1 p2 ••• pm − Ω pR2r2 T p1r ⋅⋅⋅pm − • • • 2 −Ω pRm rmTp1 p ⋅⋅⋅r 2

(4.42)

m

which is represented for the second-order tensor as ⊕

•

T = T − ΩR T +TΩR

(4.43)

Note that Ω R obeys Eq. (4.30) as known from •

•

•

Ω R∗ = R∗ R∗T = (QR )• (QR )T = (Q R + Q R ) R T QT •

•

•

= (Q R + Q R)RT QT = QQT + QΩRQT = Q(ΩR − Ω)QT

(4.44)

noting Eqs. (4.4) and (4.9). When the deformation can be ignored, all the material spins in the corotational rate described above mutually coincide as described in Eq. (2.88). Here, they are determined solely by a geometrical change of outward appearance, which is independent of individual properties of given material. On the other hand, the spin which influences on the mechanical response is the spin of substructure (microstructure) in the material. For that reason, we must adopt the corotational rate with the material spin which depends on the material property reflecting a microstructure and the deformation process. Generally speaking, the spin of the microstructure is not so large as that given by the continuum spin. An explicit form of the spin of substructure in the elastoplastic deformation will be described in Chapter 12. 4.4 Tra nsfor matio n of Material-Time Derivative

4.4

Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative

4.4 Tra nsfor matio n of Material-Time Derivative

As in the plastic constitutive equation described in the later chapters, one takes first the material-time derivative of yield condition as a scalar function and must then transform it to the corotational derivative to formulate the plastic strain rate. Here, needless to say, any scalar function and its material-derivative are observed to be identical to each other independent of coordinate systems and thus it would be

120

4 Objectivity and Corotational Rate Tensor

expected that the material-time derivative is directly transformed to the corotational derivative as shown in Eq. (4.31). On the other hand, the explicit mathematical process by which the material-time derivative of scalar function is transformed to the form that all the involved material-time derivatives are replaced to the corotational derivatives obeying Eq. (4.36) is shown below (Hashiguchi, 2007). Here, adopting the symbol

Tr(ST) ≡ S p1 p 2⋅⋅⋅pmT p1 p 2⋅⋅⋅pm

(4.45)

for arbitrary tensors S and T , the scalar-valued tensor function f ( A, B, • arbitrary tensors A, B, •

• •

••)

of

is described as shown below.

•

f ( A, B, • • • ) = Tr(

• ∂f ( A, B, • • • ) • + Tr( ∂f ( A, B, • • • ) B )+• • • A) ∂B ∂A

(4.46) Here, because f ( A, B, • • • ) is the isotropic function of the arguments A, B, • • • , it holds that f ( A, B, • • • ) = f ( A∗ , B∗ , • • • ) ⎫⎪ ⎬ • • f ( A, B, • • • ) = f ( A∗ , B∗ , • • • ) ⎪⎭

(4.47)

where the following pertains. •

f ( A∗ , B∗ , • • • ) = Tr(

∂ f ( A∗ , B∗ , • • • ) • ∗ ∂ f ( A∗ , B∗ , • • • ) • ∗ ) Tr ( A B ) +• • • + ∂B ∗ ∂A∗

c ∂ f ( A, B, • • • ) f • c f• g A∗ ) + Tr(Q d ∂ f ( A , B, • • • ) g B = Tr(Q dd g d gh ∗ ) + • • • ∂A ∂B e h e = Tr (

= Tr(

∂f ( A, B, • • • ) T • ∗ ∂f ( A, B, • • • ) T • ∗ Q a A b) + tr( Q a B b) + • • • ∂A ∂B

∂f ( A, B, • • • ) D ∂f (A, B, • • • ) D A) + T r( B) + • • • ∂A ∂B

(4.48)

Equations (4.35), (4.36) and the following equation for arbitrary tensors T and S in the same order are considered in the derivation of Eq. (4.48).

4.4 Transformation of Material-Time Derivative

121

Tr(QaTb S) = (Qp1q1 Qp2q2 • • • Qpm qmTq1q2⋅⋅⋅qm )S p1 p 2⋅⋅⋅pm

= Tq1q2⋅⋅⋅qm (Q p1q1 Q p2q2 • • • Q pm qm S p1 p 2⋅⋅⋅pm ) = Tr( T QT aS b )

(4.49)

Here, if T, S are vectors u, v , it holds that u∗ • v∗ = Q u • v∗ = u • Q T v∗ , and if they are the second-order tensors U, V , it holds that t r( U∗V∗) = t r(Q U QT V∗) = t r( U QT V∗Q) . The following expression is obtained from Eqs. (4.47)2 and (4.48). •

f ( A, B, • • • ) = Tr(

∂f ( A, B, • • • ) D ∂f ( A, B, • • • ) D A) + Tr( B) + • • • ∂A ∂B (4.50)

Therefore, the material-derivative of scalar function is transformed directly to the corotational derivative. Here, if function f involves only a single second-order tensor A , the transformation to the corotational derivative can be verified using the following equation based on Eq. (1.192), noting that tr( An ω) = tr(ωAn ) .

∂f ( A) ( Aω − ωA)} = tr{(φ0I − φ1A −φ 2 A 2 )( Aω − ωA)} = 0 tr{ ∂A (4.51) Furthermore, for the case in which two tensor variables are involved, the transformation to the corotational derivative is verified using Eq. (4.51) if they are replaced to a single variable, as in the kinematic hardening of metals and the rotational hardening of soils, which will be described in later chapters. However, in the general case for which function f involves plural tensor variables, it consists of many invariants made by them. The scalar function involving two independent tensor variables as the simplest example consists of a lot of scalar variables in Eq. (1.198) in general. Therefore, one has the inequality

Tr(

∂f ( A, B, • • • ) • ∂f ( A, B, • • • ) D A ) ≠ Tr( A) ∂A ∂A

for individual derivative terms. Then, the simple proof by individual transformation of each variable is inapplicable in this case but proofs for the transformation to the corotational derivative are necessary for all invariants including tensor operations between tensors. They usually require quite complicated calculations, while these would be the limit of itself. On the other hand, the mathematical process described

122

4 Objectivity and Corotational Rate Tensor

above gives rise to the general proof for the transformation of the material-time derivative of scalar function to the corotational derivative without such complicated processes.

4.5 Various Objective Stress Rate Tensors Various stress rate tensor with objectivity are shown below. The following Jaumann rate and Green-Naghdi (Dienes) rate of Cauchy stress tensor based on Eqs. (4.40) and (4.42) are often adopted in constitutive equations. :

•

σ ≡ σ− W σ + σ W

(4.52)

σ ≡ R(RT σ R)• RT = σ• − Ω Rσ + σΩ R ⊕

(4.53)

from which various objective stress rates are obtained as follows: •

: : • : ( ): + τ J ≡ τ (= τ − Wτ + τW ) = J Jσ = J σ J J σ J J

:

:

= σ + σ trL = σ + σ trD

(4.54)

Δ

σ ≡ Lv (σ ) = F( S )• F T = F(F −1σF −T )• F T J • −1

• −T

•

= F (F σF −T + F −1 σ F −T + F −1σ F •

−1

T •T

•

•

−1

•

• −T

)F T = F F −1 σ + σ• + σ F •

•

FT

−1

= − F F σ + σ − σ F − F = − F F σ + σ − σ (F F ) T = σ• −L σ − σLT Δ

σJ ≡

Lv (J σ ) J

(4.55) •

F S F T F (J F −1σF − T )• F T = = J J • −1

•

•

•

− F (J F −1σF −T + J F σF −T + J F −1 σ F −T + J F −1σ F T )F T = J

•

• −1 • • J = J σ + (F F )σ + σ + σ(F F −1)T

•

•

−1 −1 T • = σ − (F F )σ − σ − (F F ) + σ tr D = σ• − Lσ − σ LT + σ tr D :

= σ − Dσ − σD + σ tr D

(4.56)

4.5 Various Objective Stress Rate Tensors

123

•

τ ≡ Lv (τ ) = FSFT = F(F −1τF−T )• FT Δ

• −1

•

•

•

•

−1

•

−1

−T = F F τ + τ + σ F FT = ( − F F ) τ + τ + σ ( − F F ) T

•

= − Lτ + τ − τL

T

•

•

= −( D + W ) τ + τ − τ ( D + W )T = τ − Wτ + τW − Dτ − τD :

= τ − Dτ − τD

(4.57) •

∇

σ ≡ Lv (σ) = F −T c σ F −1 = F −T (F T σF)• F −1 •

•

= F −T (F T σF + F T σ• F + F T σ F )F −1 •

= σ +LTσ + σ L

(4.58) −1

−T

)• F T

In the above equations, the symbol Lv (T ) ≡ F (F TF (cf. Marsden and Hughes, 1983 for general definition) designates the Lie derivative, i.e. the push-forward operation FTFT of the material-time derivative of the pull-back operation of F−1TF−T of the contravariant tensor. In addition, the symbol Lv (T) ≡ F−T (FT TF)• F − 1 designates the push-forward operation F−T TF− 1 of the material-time derivative of the pull-back operation of FT TF of the covariant tensor. The objective stress rate tensors described above are listed in Table 4.1. Table 4.1. Various stress rate tensors •

:

Jaumann rate of Cauchy stress : σ ≡ σ − w σ + σ w Green - Naghdi or Dienes stress rate : σ ≡ R (R T σR )• RT = σ• − Ω R σ + σΩ R ⊕

:

: : Jaumann rate of Kirchhoff stress : τ J ≡ τ = σ + σ trD J Δ S • Oldroyd stress rate : σ J ≡ Lv (σ) = F( ) F T = F (F −1 σF −T )• F T = σ• − L σ − σLT J

Δ

Truesdell stress rate : σ ≡

Lv (J σ) J

•

=

⊗ F S FT −1 − • = F (F τF T ) F T = S + σ tr D J

:

= σ − Dσ − σD + σ tr D •

:

Truesdell rate of Kirchhoff stress : τΔ ≡ Lv (τ ) = F S F T = τ − Dτ − τD •

∇

−1 Convected or Cotter - Rivlin stress rate : σ ≡ Lv (σ ) = F −T c σ F = F −T ( F T σF )• F −1

•

= σ + LTσ + σ L

(4.59)

124

4 Objectivity and Corotational Rate Tensor

The corotational stress rate tensors obtained directly from the basic equation (4.28) of corotational tensor fulfill the following relation, noting tr(TTω) = tr(TωT) . : ⊕ t r(σ 2 )• = 2t r(σ σD ) = 2t r(σ σ ) = 2t r(σ σ )

(4.60)

In addition, the corotational rate of deviatoric tensor becomes deviatoric leading to

t r{(σ* ) : } = t r{(σ* ) ⊕ } = 0

(4.61)

Other objective stress tensors do not possess these properties. Although the objective rates are shown above for the stress rate, the objective rate must be adopted also for rates of internal variables.

4.6 Work Conjugacy Designating the volume in a specific region of material in the initial and the current configurations as V and v , respectively, the work rate W• done in this region is given from Eqs. (2.30) (2.41) (2.69) (2.73) (2.105) (3.11) (3.17) and (3.21) as follows:

，

，

，

，

，

，

•

W = ∫ tr(σ D) dv = ∫ tr (σ D ) JdV = ∫ tr( τD) dV v

v

(4.62)

v

• W = ∫ tr(σ D) dv = ∫ 1 {tr(σL) + t r(σLT )}dv v v2

= ∫ 1 {tr(σT LT ) + t r(σLT )}dv = ∫ tr(σLT ) dv v2 v •

•

•

= ∫ tr{σ (F F−1 ) T }JdV = ∫ tr( JσF −T F T ) dV = ∫ tr(ΠF T ) dV V

V

V

•

•

(4.63)

•

W = ∫ tr(σD)dv = ∫ tr(σF−T EF−1)dv = ∫ tr( F−1σF−T E)dv v

v

•

v

•

•

= ∫ tr( F −1σ F −T E)JdV = ∫ tr( J F −1σ F −T E)dV = ∫ tr(S E)dV V V V (4.64) •

• • W = ∫ tr(S E)dV = 1 ∫ tr(SC)dV = 1 ∫ tr{S(U 2 )•}dV V 2 V 2 V • • • • = 1 ∫ tr{S ( U U + U U )}dV = 1 ∫ tr{(SU + US ) U}dV = ∫ tr( Ζ U )dV V 2 V 2 V

(4.65)

4.6 Work Conjugacy

125

where Ζ ≡ 1 (SU + US ) 2

(4.66)

which is called the Biot stress tensor. The rate of work done per unit volume element in the reference configuration is given as •

•

•

•

• w = t r(τD) = t r(Π FT ) = t r(SE) = t r(SC /2) = t r(Ζ U)

(4.67)

The observation follows directly from the following equation (see Fig. 4.1). •

•

•

• = dfr • e r = Πe r • (Fe r )• = Πer • F e r = er • FT Πe r w •

•

•

= (FT Π ) rr = t r(FT Π ) = t r(Π FT )

(4.68)

noting Eqs. (2.10), (3.14) and (4.67) with dA = 1 . The unit cubic cell having the unit orthonormal outward-normal vectors e r deforms to the current cell having the outward-normal vectors er which are changed from e r , while the tractionsdf r are applied to the cell in Fig. 4.1. The work rate influencing the constitutive property is not the one done for a current unit volume but the one done for a reference volume element, i.e. a unit volume element at the initial state of deformation. The pairs of stresses and strain rates (or rates of deformation gradient) shown in Eq. (4.67) are called the work-conjugate pair. Formulation of a constitutive equation relating them pertinently is necessary for the description of finite deformation with finite rotation.

e1

− e3

e−2

e3

e2

f2

F

e2

f3 f1

−f1

e1

e−1

−f 3

−f2 Fig. 4.1 Variation of the initial unit cell due to the deformation

Chapter 5

Elastic Constitutive Equations 5 Ela stic Constitutive Equatio ns

Elastic deformation is induced by the reversible change of distances between material particles without a mutual slip between them. They therefore exhibit high stiffness. Elastic constitutive equations are classifiable into several types depending on their levels of reversibility. As preparation for the study of elastoplasticity after the next chapters, they are explained in this chapter.

5.1 Hyperelasticity The elastic material which has the strain energy function so that the one-to-one correspondence between stress and strain exists and the work done by the stress is independent of the loading path is called the hyperelastic material or Green elastic material. The constitutive equation of hyperelastic material is given as the following relation between the first Piola-Kirchhoff Π and the deformation gradient F .

Π = ρ0

∂ψ (F) ∂F

(5.1)

under the condition [∂ψ / ∂F]F=R = 0, ψ (R) = 0 , where ψ (F) is the specific strain energy (per unit mass). Substituting Eq. (5.1) into Eq. (4.63) or (4.68), the mechanical work done by a surface loading on an enclosure of certain region of the initial volume V is described as W =∫

[∫ t t r(Π F ) dt ]dV = ∫ [∫ t ρ t r( ∂ψ∂(FF) F )dt ]dV •

V

•

T

V

0

=

0

0

t•

T

∫V[∫0 ψ (F)dt ]dV = ∫V ρ0 ψ (F) dV

(5.2)

Consequently, the work done in a specific region is independent of the loading path and therefore ψ has the status of the potential energy. The energy dissipation is not induced during the stress or strain cycle. Substituting Eq. Eq. (5.1) into Eqs. (3.11), (3.17), (3.21), we obtain various expressions of the hyperelasticity by the deformation gradient as follows: K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 127–133. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

128

5 Elastic Constitutive Equations

ψ( ) ψ( ) σ = 1 ρ0 ∂ F FT = ρ ∂ F FT J

∂F

∂F

τ = ρ0 ∂ψ (F) FT ∂F

(5.3)

∂ψ (F) ∂F ∂ψ (F) S = ρ0 F −1 ∂F Π = ρ0

Furthermore, noting

∂ = ∂ ∂CPQ = ∂ ∂FrP Fr Q = ∂ (δ δ ri PA FrQ + FrPδ riδ QA ) ∂FiA ∂CPQ ∂FiA ∂CPQ ∂FiA ∂CPQ

=

∂ ∂ F = 2F ∂ F + iP ∂C AQ iQ ∂CPA iP ∂C PA

(5.4)

it holds that

∂ ψ = 2 F ∂ ψ = F ∂ψ ∂F ∂C ∂E

⎫ ⎪ ⎪ ∂ ψ = 1 − 1 ∂ ψ = 1 ∂ψ ⎪ F ⎬ ∂C 2 ∂F 2 ∂E ⎪ ∂ψ = − 1 ∂ψ = 2 ∂ψ ⎪ F ∂F ∂C ⎭⎪ ∂E

(5.5)

Then, substituting Eq. (5.5) into Eq. (5.3), the hyperelasticity is expressed in terms of the Green strain E as follows:

σ = 2 ρ F ∂ ψ (C) FT = ρ F ∂ ψ (E) FT

∂C ∂E ( ψ C ) ∂ ψ (E) T τ = 2 ρ0F ∂ FT = ρ0F F ∂C ∂E ψ (C) ψ (E) Π = 2 ρ0F ∂ = ρ0F ∂ ∂C ∂E

S = 2 ρ0

∂ ψ (C ) = ρ ∂ ψ (E ) 0 ∂C ∂E

(5.6)

5.1 Hyperelasticity

129

It holds from Eq. (1.192) for the isotropic material that

∂ ψ (E) = φ0E I + φ1E E + φ2E E 2 ∂E

(5.7)

where φ0E , φ1E , φ2E are the functions of invariants of E . Eq. (5.7) reduces to the following equation for the linear elastic material.

∂ ψ ( E) = aE ( t rE )I + 2b E E ∂E

(5.8)

where aE , b E are the material constants. The time-differentiation of Eq. (5.6) 4 leads to •

S = ρ0

∂ 2ψ (E) • E ∂E∂E

(5.9)

Substituting Eq. (2.106) and using the Truesdell rate of Kirchhoff stress Eq. (4.57), Eq. (5.9) is rewritten as 2 Δ τ = ρ0 F ∂ ψ (E) FΤ D FFΤ

∂E∂E

Here,

Δ

τ

(τΔij = ρ FiA FjBFkC FlD ∂E∂ ψ∂(EE) 2

0

AB

Δ τ in

Dkl )

CD

(5.10) is related to the Jaumann rate of Cauchy stress as Δ

:

τ = (det F ) (σ −D σ − σD + σ trD)

(5.11)

The relation between the Jaumann rate of Cauchy stress and the strain rate is given by

ρ

σ = det0 F F ∂ :

2

ψ (E )

∂E∂E

FΤ D FFΤ + D σ + σD − σ trD

(5.12)

which is expressed as :

σ = ED

(5.13)

is given by where the elastic modulus tensor E

ρ0 ∂ 2ψ (E) F F F + Σ ijkl − σijδ kl E ijkl ≡ det F FiA ∂E AB ∂ECD kC lD jB

(5.14)

130

5 Elastic Constitutive Equations

with

Σ ijkl ≡ 12 (σ ik δ jl + σ ilδ jk + σ jk δ il + σ jlδ ik ) (Σ ijkl =Σ klij =Σ jikl =Σ ijlk ) (5.15)

5.2 Cauchy Elasticity The elastic material which does not have a strain energy function but has a one-to-one correspondence between the Cauchy stress and a strain is called the Cauchy elastic material. Here, the stress tensor is given by an equation of strain tensor and thus the equation involves six strain components. The equation of six strain components fulfills the condition of complete integration leading to the strain energy function, i.e. the hyperelasticity only in special cases. Then, the work done by the stress is generally dependent on the deformation path. For that reason, an energy dissipation is induced during the stress or strain cycle. In the above-mentioned definition, the Cauchy elastic material is described as

σ = f (e)

(5.16)

Equation (5.16) reduces to the following equation by virtue of Eq. (1.202) for the isotropic material.

σ = φ0e I + φ1e e + φ2e e 2

(5.17)

where φ0e, φ1e, φ2e are functions of invariants of e . Furthermore, for an isotropic linear elastic material, Eq. (5.17) reduces to

σ = a ( t r e )I + 2b e

(5.18)

where a, b are the material constants. Limiting the infinitesimal strain leading to e ≅ ε , Eq. (5.18) results in

σ = a(t r ε )I + 2b ε

(5.19)

Here, substituting Eq. (5.19) into Eq. (3.24), the Navier’s equation is obtained as follows:

∂ 2u j ∂ 2ui • • (a + b)∇(∇ • u) + bΔu + ρb = ρ v , (a + b) ∂xj ∂x + b ∂x ∂xj + ρ bi = ρ v i j i (5.20) noting

∂u ∂u j ∂u ∂{a xk δ ij + 2b 12 ( ∂x i + )} ∂ 2u j ∂ k ∂ 2u j j ∂xi ∂ 2ui a +b +b = ∂xj ∂xi ∂xj ∂xj ∂x j ∂xi ∂xj

5.3 Hypoelasticity

131

5.3 Hypoelasticity The following material, for which the Jaumann rate of Cauchy stress is related linearly to the strain rate, is referred to as the hyperelastic material by Truesdell (1955).

σD = HD, σD ij = H ijkl Dkl

(5.21)

where H is the function of stress and internal variables. If Eq. (5.21) exhibits the isotropic rate-linearity, it is described as follows:

σD = L(t rD)I + 2GD

(5.22)

Therein, L, G are the material parameters corresponding to the Lame coefficients in Hooke’s law for the infinitesimal deformation of isotropic linear elastic material. The following relation is obtained from Eq. (5.22), and thus G is called the shear modulus.

σD 'ij = 2GDij'

(5.23)

where ( )' designates the deviatoric part described in 1.4.2. Equation (5.22) is expressed as follows:

σD = ED, σD ij = Eijkl Dkl

(5.24)

where

E ≡ LI ⊗ I + 2G I , Eijkl ≡ Lδ ijδ kl + G (δ ik δ jl + δ ilδ jk )

(5.25)

is the tangent elastic modulus tensor, which hereinafter will be simply called the elastic modulus tensor. Equation (5.25) is further rewritten as shown below.

σD = K (t rD)I + 2GD'

(5.26)

K ≡ L+ 2G 3

(5.27)

where

Using these elastic moduli, the elastic modulus tensor E is given as

⎫ ⎪⎪ ⎬ = 1 ( 1 − 1 )δ ij δ kl + 1 (δ ikδ jl + δ ilδ jk ) ⎪ 4G 3 3K 2G ⎪⎭

Eijkl = ( K − 2 G)δ ijδ kl + G (δ ik δ jl + δ ilδjk ), 3 −1

(E )ijkl

in the form of Hooke’s law.

(5.28)

132

5 Elastic Constitutive Equations

The following equation is derived from Eq. (5.26) and thus K is called the bulk modulus. (5.29) σ• = KD m

v

where ( ) m designates the mean part described 1.4.2. Furthermore, the following is obtained from Eqs. (5.23) and (5.29).

D = 1 σ• m I + 1 σD ' , Dij = 1 σ• m δ ij + 1 σD 'ij 3K 2G 3K 2G

(5.30)

from which we have

D = −3 ν σ• m I + 1 + ν σD , Dij = −3 ν σ• m δ ij + 1 +ν σD ij E E E E

(5.31)

where

− E ≡ 9 KG , ν ≡ 3K 2G 2(3K + G) 3K + G

(5.32)

The expressions below hold from Eq. (5.31) in the uniaxial loading process ( σD ij = 0 for i = j ≠ 1).

D11 = 1 σD 11 , D22 = − ν σD 11 → D22 = −ν E E D11

(5.33)

E is the ratio of the axial stress rate to the axial strain rate and is called the Young’s modulus, and ν is the ratio of the magnitude of lateral strain rate to that of axial strain rate and is called the Poisson’s ratio. The isotropic linear hypoelastic material has two independent material parameters, as described above. They are listed in Table 5.1. Table 5.1 Relationships between two independent material constants E ,ν

G,ν

E, G

E, K

G, K

L, G

μ (3L + 2G ) L+G

E

E

2(1 + ν )G

E

E

9 KG 3K + G

G

E 2(1 +ν )

G

G

3EK 9K − E

G

G

K

E 3(1 − 2ν )

2(1 +ν )G 3(1 − 2ν )

EG 3(3G − E )

K

K

L+ 2G 3

ν

ν

ν

E − 2G 2G

3K − E 6K

3K − 2G 2(3K + G )

L 2( L + G )

L

νE (1 +ν )(1 − 2ν )

2Gν 1 −ν

G ( E − 2G ) 3G − E

3K (3K − E ) 9K − E

2 K − 3G

L

5.3 Hypoelasticity

133

The following equation in which the Jaumann rate of Cauchy stress is related nonlinearly to the strain rate is called the hypoplastic material (Kolymbas and Wu, 1993)

．

σD = f (D, σ ), σD ij = fij ( Dkl , σ kl )

(5.34)

where fij is the nonlinear function of Dkl and the stress, and for rate-independent deformation it is the homogeneous function of Dkl in degree-one fulfilling fij (sDkl ) = s fij ( Dkl ) which implies (∂fij / ∂Dkl ) Dkl = fij on account of Euler’s

theorem for homogeneous function (see Appendix 3). While the tree popular elastic materials are described above, the special elastic material, called the Cosserat elastic material, is advocated by Cosserat and Cosserat (1909). The couple stress is related to the rotational strain in this material. It has been applied to the prediction of localized deformation (e.g. cf. Mindlin, 1963; Muhlhaus and Vardoulaskis, 1987).

Chapter 6

Basic Formulations for Elastoplastic Constitutive Equations 6 Basic Formulations Constitutive Equationsfor Elastoplastic

As described in Chapter 5, the elastic deformation is induced microscopically by the deformations of the material particles themselves, which returns to the initial state if the applied stress is removed. Therefore, it has a one-to-one correspondence to the stress. On the other hand, when the stress reaches the yield stress, slippage between material particles is induced and they do not return to the initial state even if the stress is removed, which leads macroscopically to the plastic deformation. For that reason, one-to-one correspondence between the stress and the strain, i.e. the stress-strain relation, observed in the elastic deformation does not hold in the elastoplastic deformation process. Therefore, one must formulate the constitutive equation as a relation between the stress rate and the strain rate in that process. This chapter presents a description of the basic concept and formulation for elastoplastic constitutive equations within the framework of conventional plasticity (Drucker, 1988) premised on the assumption that the inside of the yield surface is a purely elastic domain for the introductory to elastoplasticity. The unconventional plasticity describing the plastic strain rate induced by the rate of stress inside the yield surface will be described in subsequent chapters.

6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate 6.1 Mult iplicative Decompos itio n of Deformatio n Gradie nt

Variation in characteristics of the mechanical responses of materials arises from irreversible changes of their internal structures. The changes of internal structures are induced macroscopically by plastic deformation but they must be free from elastic deformation. Therefore, the rigorous decomposition of deformation into the elastic and the plastic deformations is necessary for description of elastoplastic deformation. Strictly speaking, the elastic deformation must be described by the purely elastic, i.e., hyperelastic constitutive equation, in which the elastic deformation returns to an initial state during a stress cycle and the work done during that cycle is zero, as described in Chapter 5. The decomposition of deformation into elastic and plastic deformations is the basic concept of elastoplasticity. On the premise of the definite decomposition of the deformation into the elastic and the plastic deformations as described above, the individual constitutive relations are formulated for these deformations in the elastoplasticity. Then, one must first decompose the current deformation into the elastic and the plastic deformations K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 135–170. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

136

6 Basic Formulations for Elastoplastic Constitutive Equations

explicitly. The elastic deformation is known from the current stress since it is related uniquely to the stress, then the plastic deformation is known as that remainder. This is realized through an explicit process: incorporating the intermediate configuration unloaded obeying the elastic constitutive relation from the current configuration, one regards deformation from the initial to the intermediate configuration as plastic deformation and the deformation from the intermediate to the current configuration as the elastic deformation. Here, note that a constitutive relation concerns with a homogeneous state of deformation. However, residual stresses exist even if a load is removed from the current configuration of a body undergoing an inhomogeneous deformation because a plastic deformation which is not removed by an unloading is induced non-homogeneously in general. Therefore, in order that all material points reach the stress-free state, individual material points must undergo different amount of de-stressing. Then, intermediate configuration can be defined only locally (point-wise). Thus, we suppose conceptually virtual intermediate configurations for each infinitesimal material elements by cutting a material into them. Based on this idea, the multiplicative (or Lee) decomposition of the deformation gradient F into the elastic deformation gradient Fe and the plastic deformation gradient F p was proposed by Eckart (1948), advanced by Kroner (1960) and further developed by Lee and Liu (1967), Lee (1969), Mandel (1972), Lubarda and Lee (1981), etc. (see Fig. 6.1) as follows:

F = FeF p

( ∂∂Xx = ∂∂Xx ∂∂XX ,

∂xi ∂xi ∂Xα = ∂ X A ∂X α ∂X A

)

(6.1)

where e ∂xi ∂x ∂X p F e = ∂X (Fiα = ∂X ), F = ∂X α

( F αp A = ∂∂XXα )

(6.2)

A

X represents the position vector of material particle in the intermediate configuration and the mapping is restricted to the class of homogeneous deformation. F is termed the Eulerian–total Lagrangian two-point tensor, Fe the Eulerian-mobile Lagrangian two-point tensor, F p the mobile Lagrangian-total Lagrangian tensor. When the stress reached the current state, different plastic deformations are induced depending on the loading path along which the current stress is realized and the elastic and plastic deformations are induced in mixed order in real unloading process. Further, a purely elastic unloading path does not necessarily exist in a real loading process. However, it is noteworthy that the real existence of purely elastic unloading process is not required since the intermediate configuration is known by the elastic constitutive relation. It holds from Eq. (6.1) with Eq. (1.22) that

detF = det(F eF p ) = detF e detF p

(6.3)

On the other hand, presuming that line-elements ΔX , ΔX , ΔX in the initial state change to Δx a , Δxb , Δxc in the current state and to ΔXa , ΔXb , ΔXc in a

b

c

6.1 Multiplicative Decomposition of Deformation Gradient

137

X 3 , X 3 , x3

dx

F

x

Fe

X dX

dX

F

X

p

0

X 2 , X 2 , x2

X1 , X1 , x1 Fig. 6.1 Multiplicative decomposition of deformation gradient

the intermediate state, and denoting the initial, current and intermediate volume elements as V , v and V , respectively, it holds that

Δv = (Δxa × Δxb ) • Δxc = ε ijk Δxia Δxjb Δxkc = detΔx Δx1a Δx1b Δx1c

F1R ΔX Ra F1R ΔX Rb F1R ΔX Rc

a b c F11 F12 F13 ΔX 1 ΔX 1 ΔX 1

= Δx2a Δxb2 Δx2c = F2 R ΔX Ra F2 R ΔX Rb F2 R ΔX Rc = F21 F22 F23 ΔX 2a ΔX 2b ΔX 2c = J ΔV Δx3a Δx3b Δx3c

F3 R ΔX Rc F3 R ΔX Rb F3 R ΔX Rc

Δx1a Δx1b Δx1c

F1eR ΔX Ra F1Re ΔX Rb F1eR ΔX Rc

F31 F32 F33 ΔX 3a ΔX 3b ΔX 3c F11e F12e F13e ΔX 1a ΔX 1b ΔX1c

Δv = Δx2a Δxb2 Δx2c = F2eR ΔX Ra F2eR ΔX Rb F2eR ΔX Rc = F21e F22e F23e ΔX 2a ΔX 2b ΔX 2c = J e ΔV Δx3a Δx3b Δx3c ΔX 1a ΔX 1b ΔX 1c

F3eR ΔX Ra F3eR ΔX Rb F3eR ΔX Rc F1Rp ΔX Ra F1Rp ΔX Rb F1Rp ΔX Rc

F31e F32e F33e ΔX 3a ΔX 3b ΔX 3c p p p F11 F12 F13 ΔX 1a ΔX 1b ΔX 1c

ΔV = ΔX 2a ΔX 2b ΔX 2c = F2pR ΔX Ra F2Rp ΔX Rb F2pR ΔX Rc = F21p F22p F23p ΔX 2a ΔX 2b ΔX 2c = J p ΔV ΔX 2a ΔX 3b ΔX 3c

F3pR ΔX Ra F3Rp ΔX Rb F3pR ΔX Rc

F31 F32 F33 ΔX 3a ΔX 3b ΔX 3c p

p

p

It obtained from these equations that

J = J eJ p

(6.4)

138

6 Basic Formulations for Elastoplastic Constitutive Equations

where

J ≡ detF = Δv , J e ≡ detFe = Δv , J p ≡ detF p = ΔV ΔV ΔV ΔV

(6.5)

Equation (6.4) engenders the additive decomposition of the logarithmic volumetric strain ε v into the elastic logarithmic volumetric strain ε ve and the plastic logarithmic volumetric strain ε vp , i.e.

ε v = ε ve + ε vp

(6.6)

where

ε v ≡ ln J = ln Vv , ε ve ≡ lnJ e = ln v , ε vp ≡ lnJ p = ln V

V

V

(6.7)

for a homogeneous deformation. It holds from Eqs. (6.7) that •

•

•e

•

•

•p

•

ε• v = JJ = vv , ε• ev = J e = vv − V , ε• v = J p = V J J V V p

(6.8)

Substitutions of Eqs. (6.1) into the velocity gradient L in Eq. (2.69) leads to • •p −1 −1 L = F e F e + F e F F p F e −1 = Le + L p

(6.9)

where

⎫ ⎪ ⎪ • • ⎪ ⎪ • −1 ∂ X ∂X = ∂ X ⎬ Lp ≡ F p F p = ∂X ∂X ∂X ⎪ ⎪ • p −1 L ≡ F e Lp F e = ∂x ∂ X ∂X ⎪ ∂X ∂X ∂x ⎭⎪ •e −1 ∂x • ∂X Le ≡ F F e = ( ) ∂x ∂X

(6.10)

p can be decomposed additively into the symmetric and Furthermore, Le , Lp , L skew-symmetric parts as follows: e Le = D + W e ⎫ p p⎪ Lp = D + W ⎬ ⎪ p p +W p⎭ L = D

(6.11)

6.1 Multiplicative Decomposition of Deformation Gradient

139

where ⎫ • De = (Le ) s = (F e F e −1 ) s , ⎪ ⎬ • W e = (Le ) a = (F e F e −1 ) a ⎪⎭ • ⎫ • −1 D p = (Lp ) s = ( F p F p ) s = ( ∂ X )s , ⎪ ⎪ ∂X ⎬ • ⎪ • −1 W p = (Lp ) a = ( F p F p ) a = ( ∂ X )a ⎪ ∂X ⎭

(6.12)

(6.13)

p = (L p ) = (F e Lp F e −1 ) = (F e D p F e −1 ) s + (F e W p F e −1 ) s , ⎫⎪ D s s ⎬ 6.14) −1 −1 p p p p e e p e −1 e e e ) ) W = (L )a = (F L F ) a = (F D F a + (F W F a ⎪⎭ where ( )s and ( )a denotes the symmetric and the skew(anti)-symmetric parts, respectively. In the similar manner as in Eq. (2.98) it holds that •

•

( d X • δ X )• = d X • δ X + d X • δ X •

•

•

•

T = ∂ X dX • δ X + dX • ∂ X δ X = [{∂ X + ( ∂ X ) }dX] • δ X ∂X ∂X ∂X ∂X

e −T e −1 = 2 D p d X • δ X ( = 2F D p F dx • δ x )

(6.15) The strain rate D and the spin W are additively decomposed by substituting Eq. (6.9) into Eqs. (2.73) and (2.74) as follows:

p = De + (F e D p F e −1 ) s + (F e W p F e −1 ) s D = (Le ) s + (L p ) s = De + D

½° ¾ p = W e + ( F e D p F e −1 ) a + ( F e W p F e −1 ) a ° W = (Le )a + (L p )a = W e + W ¿ (6.16)

Furthermore, the following equations hold for the volumetric strain rate from Eqs. (2.73), (6.10), (6.12)-(6.14).

140

6 Basic Formulations for Elastoplastic Constitutive Equations

⎫ ⎪ ⎪ ⎪ • • • ⎪ • p ( ) ∂ ∂ ∂ X X X • ∂ 1 p p 2 3 X V i Dv ≡ t rD = t r ∂ X = = = = ε v ,⎪ ⎪ ∂X ∂X i V ∂X 1∂X 2 ∂X 3 ⎬ • • • ⎪ p p = t r( ∂ x ∂ X ∂X ) = t r ∂ X = V = ε• v = D p ⎪ D vp ≡ t rD v ∂X ∂X ∂ x ∂X V ⎪ ⎪ • • ⎪ e • p v V e e = − =εv Dv ≡ t rD = t rD − t rD ⎪ v V ⎭ • • (∂x1∂x2 ∂x3 )• v• • Dv ≡ t rD = t r ∂ x = ∂ xi = = v = ε v, ∂x ∂xi ∂x1∂x2 ∂x3

(6.17)

from which the additive decomposition holds exactly for the volumetric strain rate, i.e.

Dv = Dve + Dvp

(6.18)

Now, let Q (t ) be a time-dependent rotation superimposed on the current configuration, by which deformation gradient is given as F∗ = QF

(6.19)

Here, let it be postulated that the initial and the intermediate configurations are fixed so that they are not affected by the superposition of rigid-body rotation (Dashner, 1986). Then, it holds that

F∗ = Fe∗F p∗; F e∗ = QFe , F p∗ = F p

(6.20)

The elastic part of the right Cauchy-Green tensor in Eq. (2.30) defined in the intermediate configuration, denoted Ce , is given by Eq. (6.20) as

Ce∗ = Fe∗T Fe∗ = (QFe )T QFe = FeT Fe ≡ Ce

(6.21)

Therefore, the hyperelastic constitutive equation described by Eq. (5.6) 4 , i.e. S = 2 ρ0 ∂ ψ (Ce ) / ∂ Ce holds independent of the superposition of rigid-body rotation, noting S∗ = S in Eq (4.19), where S is the second Piola-Kirchhoff stress defined in the intermediate configuration ( F is replaced to F e ), the explicit expression of which will be shown in 6.7. Eq. (6.20) imposes the basis describing elastic deformation, i.e. the intermediate configuration to be unaffected by the rigid-body rotation. Thus, the multiplicative decomposition is required to obey Eq. (6.20) for the fulfillment of objectivity of elastic constitutive equation.

6.1 Multiplicative Decomposition of Deformation Gradient

141

Now, consider the influence of the rigid-body rotation on the main rate variables for the multiplicative decomposition fulfilling Eq. (6.20). The substitution of Eq. (6.20) into Eq. (6.10) leads to

Le∗ = (F e∗ ) • F e∗ = (QF e ) • (QF e ) −1 −1

•

•

e = (Q Fe + Q F )Fe −1Q −1 • • = Q F e Fe −1 QT + Q QT

= Q(Le − Ω)QT

(6.22)

where Ω is the spin of the base {e∗i } rotating with a material as shown in Eq. (4.12). Then, we have

⎫ ⎪⎪ ⎬ e 1 e∗ e∗T T e W ∗ = 2 (L − L ) = Q( W − Ω)Q ⎪ ⎪⎭

D e∗ = 1 (Le∗ + Le∗T ) = QDeQT 2

(6.23)

Next, substituting Eq. (6.20) into Eq.(6.10), one has ⎫ L p∗ = L p ⎪ p T⎬ p∗ e∗ p∗ e∗−1 (Q e ) p (Q e ) −1 Q e p e −1QT Q = F L = F LF = QL L =F L F F ⎪⎭ (6.24) Then, It holds from Eq. (6.13) and (6.14) that

D p ∗ = D p , W p∗ = W p

(6.25)

p∗ = QD p QT , W p∗ = Q W p QT D

(6.26)

Consequently, the elastic strain rate De and the plastic strain rate D p indicate the objectivity. Apart from the exact formulation conforming to the hyperelasticity, let it be postulated that the unloading process to the intermediate configuration is a purely elastic deformation (Lee, 1969) and the elastic deformation is infinitesimal (Dafalias, 1985). Then, it holds in the polar decomposition of elastic deformation gradient Fe = V e R e that

F e = V e ≅ I, R e = I

(6.27)

142

6 Basic Formulations for Elastoplastic Constitutive Equations

In this case, substituting Eq. (6.27) into Eq. (6.16), the following equation holds.

⎫ ⎪ ⎬ • p = (F• p F p −1)a ) ⎪ (We = ( V e Ve −1 )a , W p = W ⎭

• • p p = ( F• p F p −1) ) D = De + D p (De = (V e )s = V e , D = D s

W=

We

+ Wp

(6.28)

with an error of first order in small elastic strain. Here, note that the rigid-body rotation is attributed to the variation of F p in direct opposition to the above-mentioned Dashner’s (1986) notion in Eq. (6.20) and that W p differs substantially from the so-called plastic spin (Dafalias, 1985) as will be described in Chapter 12. The additive decomposition of strain rate D into the elastic strain rate De and the plastic strain rate D p in Eq. (6.28) is adopted in the formulation of elastoplastic constitutive equations with the hypoelasticity for De in the subsequent sections.

6.2 Conventional Elastoplastic Constitutive Equations First, let the elastic strain rate be given by the following hypoelastic constitutive equation from Eq. (5.24). D

De = E −1 σ

(6.29)

Next, the plastic strain rate must be formulated. As described in the beginning, let the conventional elastoplastic constitutive equations (Drucker, 1988) be described below. Now, to formulate the plastic strain rate, consider first the following isotropic yield condition exhibiting the isotropic hardening/softening.

f (σ ) = F ( H )

(6.30)

where F is the function of the isotropic hardening variable H and is called the hardening function. Hereinafter, assume that the yield stress function f (σ) is the function of stress invariants and is the homogeneous function of σ in degree-one. Therefore, it holds that (6.31) f (s σ) = s f (σ) for the arbitrary positive scalar

s

and ∂ f (σ ) tr( ∂σ σ) = f (σ )

(6.32)

for sake of Euler’s theorem for homogeneous function in degree-one (see Appendix 3). Then, it holds from Eqs. (6.30) and (6.32) that

6.2 Conventional Elastoplastic Constitutive Equations

∂f (σ) = ∂σ

143

σ tr ∂f ( ) σ f (σ) ∂σ N= N= F N tr(Nσ) tr(Nσ) tr(N σ)

(

)

(6.33)

where N is the normalized outward-normal of the yield surface (see Appendix 4), which is described as N≡

∂f (σ) ∂σ

∂f (σ) ∂σ

( N = 1)

(6.34)

Here, the yield surface in Eq. (6.30) retains the similar shape and orientation with respect to the origin of stress space by virtue of homogeneity of function f (σ) . Taking the material-time derivative of Eq. (6.30) and considering the transformation to the corotational derivative (Hashiguchi, 2007) described in Chapter 4, one has the consistency condition:

tr( ∂

f (σ ) D σ) = F' H• ∂σ

(6.35)

where

F ' ≡ dF / d H ( ≥ 0)

(6.36)

•

H is assumed to be the function of stress, internal variables and the plastic strain rate, i.e. •

H = h (σ, H i ; D p )

(6.37)

signifies internal variables collectively. Here, note that h has to be the homogeneous function of D p in degree-one fulfilling h (σ, H i ; s D p ) = s h (σ, H i ; D p ) for the rate-independent deformation behavior, while, needless to

H i (i = 1, 2,

• •• )

p

say, h is a nonlinear equation of Dij in general as seen in metals described later. The substitution of Eq. (6.33) into the consistency condition (6.35) leads to • tr (N σD ) = F' tr(N σ ) H F

(6.38)

Further, assume the plastic flow rule

D p = λ M (||M|| ≠ 1)

(6.39)

where λ is the positive proportionality factor and M is the second-order tensor function of stress and internal variables. Substituting the flow rule (6.39) into the consistency condition (6.38) with Eq. (6.37), one has

144

6 Basic Formulations for Elastoplastic Constitutive Equations

tr (N σD ) = F' tr(Nσ)λ h ( σ, Hi ; M) F from which it is derived that D

D

σ σ λ = tr(Np ) , D p = tr(Np ) M M M

(6.40)

where M p is called the plastic modulus and is given by

Mp≡

F ′ h( σ, H ; M) tr(Nσ) i F

(6.41)

Substituting Eq. (6.29), (6.40) into Eq. (6.23)1, the strain rate is given by D = E −1 σD +

D

tr(N σ ) M Mp

(6.42)

The proportionality factor described in terms of strain rate, denoted by Λ instead of λ , in the flow rule (6.39) is given from Eq. (6.42) as follows:

Λ=

tr(N ED) M p + tr( NEM)

(6.43)

Actually, Eq. (6.43) can be derived directly by substituting the flow rule (6.39) into the following consistency condition in terms of the strain rate obtained by substituting Eqs. (6.23), (6.29), (6.33) and (6.37) into the consistency condition (6.38) in terms of stress rate.

tr{NE(D − Dp )} = F' tr(Nσ)λ h (σ, Hi ; M) F

(6.44)

The inverse relation of Eq. (6.42) is given by using Eq. (6.43) as follows:

σD = ED −

tr(NED) ⊗ EN ) D EM = (E − EM M p + tr( NEM) M p + tr( NEM)

σD ij = Eijrs Drs −

N pq E pqmn Dmn Eijrs M rs M + N ab Eabcd M cd p

(6.45)

Using the plastic relaxation modulus tensor K pr ≡

Kijklr ≡ p

EM ⊗ EN , M p + tr( NEM)

Eijrs M rs N pq E pqkl M p + N ab Eabcd M cd

(6.46)

6.2 Conventional Elastoplastic Constitutive Equations

145

the plastic relaxation stress rate σD p is described as

σD p ≡ ED p = K pr D Furthermore, using the elastoplastic stiffness modulus tensor Kep ≡ E − K pr = E −

EM ⊗ EN M p + tr( NEM) ,

e pr K ijkl = Eijkl − ≡ Eijkl − K ijkl p

Eijrs M rs N pq E pqkl M p + N ab Eabcd M cd

(6.47)

the stress rate can be described as

σD = K ep D As known from Eqs. (6.46) and (6.47), the plastic relaxation modulus tensor and the elastoplastic stiffness modulus tensor are the asymmetric tensors, i.e.

K p ≠ K p T , Kep ≠ Kep T in general. As described in the next section, however, these stiffness modulus tensors become symmetric when the direction of plastic strain rate is chosen to be the outward-normal of yield surface, i.e. M = N . In this case the plastic flow rule is called the associated flow rule or the normality rule because the direction of plastic strain rate is given using the yield function as the plastic potential function. Hereinafter, the explicit equations of the above-mentioned constitutive equations adopting the associated flow rule are shown collectively as a support for the explanation of subsequent formulations. D

σ D p = λ N , D p = tr(Np ) N M Mp≡

F ′ h ( σ, H i ; N) tr(Nσ) F

D = E −1 σD +

D

t r(N σ ) N Mp

(6.48)

(6.49)

(6.50)

from which we have

Λ=

tr(NED) M + tr( NEN)

tr(NED) ⊗ EN ) D EN = (E − EN M p + tr( NEN) tr( NEN)

σD = ED − M p +

(6.51)

p

(6.52)

146

6 Basic Formulations for Elastoplastic Constitutive Equations

K pr ≡

EN ⊗ EN M p + t r( NEN)

，K

ep

≡ E−

EN ⊗ EN M p + tr( NEN)

(6.53)

The elastoplastic constitutive equations for isotropic materials are described above. The constitutive equation of metals is shown below, which has made an important contribution to the development of elastoplasticity. The following von Mises yield condition with the associated flow rule is assumed for metals.

f (σ) = σ e , σ e ≡ 3 ||σ' || 2 F ( H ) = F (ε p e ), H = ε p e ≡ ∫ M=N In the monotonic simple tension ( σ ij = 0 except for

⎫ ⎪ ⎪ ⎪⎪ 2 || p || dt ⎬ D 3 ⎪ ⎪ ⎪ ⎪⎭

(6.54)

i = j = 1 , D22p = D33p = − D11p / 2

p (trD p =0) , Dij = 0 (i ≠ j ) ) it holds that

⎫ ⎪ p 2 p2 p p⎬ 2 − dt = ∫ D11 dt = ε11 ⎪ 3 D11 + 2( D11 / 2) ⎭

σ e = 3/2 (σ 11 − σ 11 / 3) + 2(0 − σ 11 / 3) = σ 11 ε pe ≡ ∫

(6.55)

Then, σ e and ε p e coincide with the axial stress and the axial plastic strain in that loading and thus are called the equivalent stress and the equivalent (or accumulated) plastic strain, respectively. Substituting Eq. (6.54) into Eqs. (6.50)-(6.52) and using the relations

∂f (σ) = ∂σ

3 σ' , N = σ' , tr(Nσ ) = ||σ || ' 2 ||σ' || ||σ'||

σ• e = 3 tr ( σ' σD ' ) = 3 tr(N σD ) 2 ||σ || 2

'

D p = D p' •

H ( σ, H i ; D p ) = M p = F' F

2 || p ||, h( σ, H i ; N) = D 3

2 ||σ || = 2 F 3 ' 3 '

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (6.56) ⎪ 2 || || = 2 ⎪ N 3 3 ⎪ ⎪ ⎪ ⎪ ⎭

6.2 Conventional Elastoplastic Constitutive Equations

147

the constitutive equation of the isotropic Mises material is given as follows:

2 •e D 3σ σ N tr( ) σ' = E −1 σD + 3 1 σ• e σ N = E −1 σD + D = E −1 σD + 2 F' σ e ' 2 F || σ' || 2F ' ' 3 3 (6.57) which is called the Prandtl-Reuss equation. The plastic work rate of this material is described as

t r(σ D p ) = t r (σ λ σ' ) = ||σ' || λ = || σ' |||| D p || = σ e ε• pe = F ε• pe ||σ' ||

(6.58)

which is the product of the hardening function and the equivalent plastic strain and thus the hardening attributable to the equivalent plastic strain is also called the work hardening. The traction t acting on the plane having the normalized outward-normal vector em inclined identically to the principal stress direction, called the octahedral plane, is given by

t = σ em = (σⅠeⅠ ⊗ eⅠ + σⅡeⅡ ⊗ eⅡ + σ ⅢeⅢ ⊗ eⅢ )( 1 eⅠ + 1 eⅡ + 1 eⅢ ) 3 3 3 = 1 (σⅠeⅠ + σⅡeⅡ + σ ⅢeⅢ ) 3

(6.59)

noting Eq. (3.7). Then, the normal component σ oct and the tangential component

τ oct

of the traction t is given as

σ oct ≡ t • em = 1 (σⅠeⅠ + σⅡeⅡ + σ ⅢeⅢ ) • ( 1 eⅠ + 1 eⅡ + 1 eⅢ ) 3

3

3

= 1 (σⅠ + σⅡ + σ Ⅲ ) = σ m 3

3

(6.60)

2 τ oct ≡ || t ||2 −σ oct

= 1 (σⅠ2 + σⅡ2 + σ Ⅲ2 ) − 1 (σⅠ + σⅡ + σ Ⅲ )2 9 3 =

2 {σ 2 + σ 2 + σ 2 − (σ σ + σ σ + σ σ )} Ⅲ Ⅰ Ⅰ Ⅱ Ⅱ Ⅲ Ⅱ Ⅲ 9 Ⅰ

= 1 (σⅠ − σⅡ ) 2 + (σⅡ − σ Ⅲ ) 2 + (σ Ⅲ − σⅠ) 2 = 3

2 J 3 2

(6.61)

148

6 Basic Formulations for Elastoplastic Constitutive Equations

where τ oct is called the octahedral shear stress and J 2 is given by choosing the deviatoric Cauchy tensor σ' as the second invariant of the deviatoric tensor T' in Eq. (1.150), i.e.

J 2 ≡ 1 ||σ' || 2 = 1 tr σ' 2 = 1 σ 'rsσ 'sr 2 2 2 1 ' 2 + σ 2 + σ 2 ) + σ 2+ σ 2 + σ 2 = 2 (σ 11 '22 '33 '12 '23 '31

1 = 2 (σⅠ' 2 + σⅡ ' 2 + σ Ⅲ' 2 ) = 16 {(σⅠ − σⅡ) 2 + (σⅡ − σ Ⅲ ) 2 + (σ Ⅲ − σⅠ) 2} (6.62) It is interpreted from Eqs. (6.54), (6.61) and (6.62) for the Mises yield condition that the yielding is induced when the octahedral shear stress reaches a certain value. Eqs. (6.60) and (6.61) are also derived as the components in the directions I m ( = 3e m ) and 3 t' , i.e. Tm and || T' || / 3 of Tm and T' , respectively., regarding T as σ in Eq. (1.210).

6.3 Loading Criterion The judgment of whether or not the plastic strain rate is induced for a given incremental loading is required for the elastoplastic deformation analysis. The criterion for this judgment is called the loading criterion. In what follows, this criterion is formulated (Hashiguchi, 2000). 1. It is required that

λ=Λ>0 p in the loading (plastic deformation) process D ≠ 0 . 2. It holds that

(6.63)

D (6.64) tr( N σ) ≤ 0 in the unloading (elastic deformation) process D p = 0 . Further, substituting D e D = De leading to tr(NED) = tr(NED ) = tr(N σ) into Eq. (6.43), Λ is described in this process as

Λ=

D

t r( N σ ) M + t r( NEM ) . p

(6.65)

in this process. 3. The plastic modulus M p takes both positive and negative signs in general. On the other hand, noting that the elastic modulus E is the positive definite tensor and thus it holds that t r(NEM ) M p in general and postulating that the plastic relaxation does not proceed infinitely, let the following inequality be assumed.

6.3 Loading Criterion

149

M p + t r(NEM) > 0 .

(6.66)

pr → ∞ is illusThe ultimate state fulfilling M p + t r( NEM ) → 0 leading to Kijkl trated for the uniaxial loading process in Fig. 6.2. Then, in the unloading process D p = 0 , the following inequalities hold from Eqs. (6.40) and (6.63)-(6.66), depending on the sign of the plastic modulus M p leading to the hardening, perfectly-plastic and softening state.

λ ≤ 0 and Λ ≤ 0 when M p > 0

⎫ ⎪ λ → −∞ or indeterminate and Λ ≤ 0 when M = 0⎬ (6.67) ⎪ λ ≥ 0 and Λ ≤ 0 when M p < 0 ⎭ p

Consequently, the sign of λ at the moment of unloading from the state m p ≤ 0 is not necessarily negative. On the other hand, Λ is necessarily negative in the unloading process. Thus, the distinction between a loading and an unloading process cannot be judged by the sign of λ but it can be done by that of Λ . Therefore, the loading criterion is given as either

D p ≠ 0 : f (σ) = F ( H ) and Λ > 0 ⎪⎫ ⎬ D p = 0 : otherwise ⎪⎭

σ

tr ( N ) > 0 t r ( NED) > 0

}

p

M =0

tr ( N ) = 0 t r ( NED) > 0

M p< 0

Mp> 0

tr ( N ) < 0 t r ( NED) < 0

}

}

tr ( N ) < 0 t r ( NED) < 0

}

tr ( N ) < 0 t r ( NED) < 0

tr ( N ) < 0 t r ( NED) > 0

}

0

D Fig. 6.2 Signs of tr(N σ ) and t r(NED) in uniaxial loading

}

M p + t r ( NEM) → 0 pr Kijkl → ∞

ε

(6.68)

150

6 Basic Formulations for Elastoplastic Constitutive Equations

or

D p ≠ 0 : f (σ) = F ( H ) and tr(NED) > 0 ⎫⎪ ⎬ D p = 0 : otherwise ⎪⎭

(6.69)

(Hill, 1967, 1983) in lieu of Eq. (6.66). Limiting to the hardening process, Eq. (6.69) leads to ⎫ D p ≠ 0 : f (σ) = F ( H ) and tr(N σD ) > 0 ⎪ ⎬ ⎪⎭ D p = 0 : otherwise

Denoting the infinitesimal total strain and plastic strain by ε and tively, the elastic strain

(6.70)

ε p,

εe = ε − ε p

respec(6.71)

is related uniquely to the stress. Then, substituting Eq. (6.71) in to Eq. (6.30), the yield surface can be described in the strain space as follows:

g (εe ) = F ( H ) or g (ε − ε p ) = F ( H )

(6.72)

setting g (ε e ) ≡ f (σ (ε e)) . Then, one has

∂f (σ ) ∂g (ε e ) ∂ε e ∂g (ε e ) D e ∂g (ε − ε p ) E −1 ∂g (ε − ε p ) E −1 = = = ( = ∂σ ∂ε ∂ε e ∂ σ ∂ε e σD ∂ ε −εp)

(6.73)

The substitution of Eq. (6.73) into tr(NED) > 0 in Eq. (6.69) leads to tr( ∂

g(ε − ε p ) D) > 0 ∂ε

(6.74)

Therefore, it can be interpreted that loading occurs when the strain rate is directed the outward-normal of the yield surface in the strain space. In elastoplastic deformation analysis, suppose to calculate first σD and D by either of the elastic or the elastoplastic constitutive equation. Then, check the sign of t r(NED) . If the sign conflicts with the loading criterion, it is required to recalculate them using other constitutive equation. Here, it would be efficient to calculate first by the elastoplastic constitutive equation since the monotonic loading process in which the elastoplastic deformation process continues is seen often in practical engineering problems.

6.4 Associated Flow Rule

151

6.4 Associated Flow Rule The associated flow rule holds under some assumptions for a wide range of materials. Some mechanical interpretations for the associated flow rule are delineated in this section.

6.4.1

Positivity of Second-Order Plastic Work Rate: Prager’s Interpretation

Prager (1949) reported that the associated flow rule must hold to fulfill the positivity of the second-order plastic work rate. i.e.

w p ≡ t r(σD D p) / 2 ≥ 0

(6.75)

However, Eq. (6.75) holds only for the hardening process in which the stress rate is directed the outwards of the yield surface.

6.4.2

Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis

Drucker (1951) postulated “the work done during the stress cycle by the external agency is positive”. It is described mathematically as follows:

vσ∫ t r{( σ −σ

0

)Ddt } ≥ 0

(6.76)

where σ 0 denotes the initial stress. The following inequality is obtained from Eq. (6.76) under the assumption that the inside of the yield surface is elastic domain (see Fig. 6.3). (6.77) ( σ y − σ 0 )D p ≥ 0 where σ y designates the stress on the yield surface in which the plastic strain rate

D p is induced. The followings should hold in order to fulfill Eq. (6.77). 1) The plastic strain rate is directed outward-normal of the yield surface. Then, the associated flow rule must hold, provided that the direction of plastic strain rate is determined solely by the current stress and internal variable but independent of stress rate. 2) In this occasion the yield surface has to be the convex surface (see Fig. 6.4). The result 1) is called the associated flow rule or the normality rule and the result 2) is called the convexity of yield surface.

152

6 Basic Formulations for Elastoplastic Constitutive Equations

σ σy

σ0

t r{(

y

−

0

) D pdt } ≥ 0

D p dt

ε

0

Fig. 6.3 Positive work done by external agency in Drucker’s (1951) postulate (Illustration in uniaxial loading process)

Dp

Yield surface

y

00

00 00

00

00

Dp

Elastic Elast ic region region 00

.

yy

σij , ε ijp

0

t r {(

Normality rule

y

−

0

) D p }< 0

Concave yield surface : Violation of Drucker (1951 )’s postulate .

Fig. 6.4 Associated flow (normality) rule and convexity of yield surface based on the Drucker’s (1951) postulate

6.4.3 Positivity of Second-Order Plastic Relaxation Work Rate Ilyushin (1961) postulated that “the work done during the strain cycle is positive”. Limiting to the infinitesimal deformation process, it leads to the postulate “the

6.4 Associated Flow Rule

153

second-order work rate w is smaller than the second-order elastic stress work rate wse calculated by presuming that the strain rate is induced elastically” or “the second-order plastic relaxation work rate w pr , i.e. the work rate done during the infinitesimal strain cycle is positive” (Hill, 1968; Hashiguchi, 1993a; Petryk, 1993; see Fig. 6.5). It is described mathematically as follows:

w ≤ wse , w pr ≥ 0

(6.78)

where

⎫ ⎪ ⎪ wse ≡ t r(σD e D) / 2 = tr( DED) / 2 ⎬ ⎪ p w pr ≡ tr(σD D) / 2 = tr(D p ED) / 2 ⎪ ⎭ w ≡ t r(σD D) / 2 = tr( De ED) / 2

(6.79)

with

σD e ≡ ED, σD p ≡ ED p

(6.80)

σD e and σD p are called the elastic stress rate and the plastic relaxation stress rate, respectively. It should be noted that the associated flow rule must hold for Eq. (6.78) 2 to meet the loading condition in Eq. (6.68).

6.4.4 Comparison of Interpretations for Associated Flow Rule Prager’s (1949) interpretation of the associated flow rule is concerned only with hardening materials as described previously. On the other hand, the interpretation of the positivity of work done by the external agency, i.e. the additional stress during the stress cycle by Drucker (1951) and the positivity of the second-order plastic relaxation work rate are based on postulates of the dissipation energy of materials independent of hardening behavior. Here, Drucker’s (1951) postulate is related to the stress cycle but the postulate of the second-order plastic relaxation work rate is related with the infinitesimal strain cycle. Now, compare below the pertinence of these postulates. (1) The strain cycle can be realized always. However, the stress cycle cannot be made in the softening state in which the stress cannot be returned to the initial state if the plastic strain rate is induced. It is based on the fact that any deformation can be given, but a stress cannot be given arbitrarily to materials since strength of materials is limited.

154

6 Basic Formulations for Elastoplastic Constitutive Equations

σ d dσ σ pp dσ σ

σ

E

ee

1

dσ

a

w pprr e

b f

1 E

c

0

ε

dε p dε e dε

Ha rdening process ( M p > 0 )

σ d

dσσ pp

E

dσσ ee

1

a

σ

f

dσσ w pprr 1

e

b

E

c

0

dε p dε e Softening process ( M p < 0 )

ε

dε

Fig. 6.5 Positiveness of second order work rate (Illustration in uniaxial loading)

6.4 Associated Flow Rule

155

(2) Limiting to the infinitesimal cycles, consider the stress and strain cycles. The second-order work increment done during the infinitesimal stress cycle is given by t r (σD dt D pdt ) / 2 ( Δ abe in the hardening process in Fig. 6.5). On the other hand, the additional work increment tr(D pdt ED pdt ) / 2 ( Δaec in the hardening process in Fig. 6.5) must be done to close the strain cycle, whilst tr( D pdt ED pdt ) / 2 > 0 holds because of the positive-definiteness of the elastic modulus tensor E. Therefore, the work done during the infinitesimal stress cycle is far smaller than the work during the infinitesimal strain cycle. In other words, Drucker’s (1951) postulate holds for the materials fulfilling a more restricted condition, i.e., more particular materials than the materials fulfilling the positivity of the second-order plastic relaxation work rate. (3) The strain (increment) has one-to-one correspondence to the displacement (increment) induced in the material. Therefore, the configuration of material returns to the initial state only if the strain in any definition returns to the initial value. In other words, if the strain cycle in any definition of strain (logarithmic strain, nominal strain for instance) closes, the strain cycle in the other definition also closes. On the other hand, the configuration in the end of stress cycle differs from the initial configuration depending on the loading path chosen during the cycle and on the definition of stress (Cauchy stress, nominal stress for instance). Eventually, the strain cycle possesses the objectivity, but the stress cycle does not possess it. (4) The assumption that the interior of the yield surface is the purely elastic domain is adopted in Drucker’s postulate. On the other hand, it is not required by the postulate of the positivity of second-order plastic relaxation work rate, which holds on the quite natural premise that the purely elastic deformation is induced at the moment of unloading. Eventually, it can be stated that postulate of the positivity of second-order plastic relaxation work rate is more general than Drucker’s postulate. However, even the former is based on the premise that the direction of the plastic strain rate is dependent on the normal component but independent of the tangential component of stress rate to the yield surface. It is observed in the test result that the inelastic deformation is induced even by the tangential component. The inelastic strain rate induced by the tangential component will be described in 6.6. Regarding the case in which the plastic strain rate is directed the outward-normal of convex yield surface, i.e. the case in which the associated flow rule holds, the plastic work rate done by the actual stress σ y on the convex yield surface is greater than that done by any statically-admissible stress σ* as depicted in Fig. 6.6. It is called the principle of maximum plastic work. It is beneficial for the formulation of variational principles.

156

6 Basic Formulations for Elastoplastic Constitutive Equations

N

ıy

ı* σ ij

0

Yield surface Fig. 6.6 Principle of maximum plastic work

6.5 Anisotropy The plastic strain rate described in 6.2 concerns the yield condition with the function f involving only the stress invariants. Therefore, it is limited to the materials exhibiting the isotropy in the plastic deformation behavior. In what follows, first the isotropy in constitutive equation is defined. Then, the plastic strain rate extended to the anisotropy will be explained in this section.

6.5.1 Definition of Isotropy An isotropic material is defined as one exhibiting identical mechanical response that is independent of the chosen direction of material element or of the coordinate system by which the response is described. Here, the input/output variables are the stress rate and the strain rate in the irreversible deformation. The rate-type constitutive equation in a rate form is described in general as follows: D (6.81) )

f (σ, σ, Hi , D = 0

When the following equation holds by giving coordinate transformations only for stress (rate) and strain rate tensors in the function f , it can be stated that Eq. (6.81) describes the constitutive equation of isotropic material. D D (6.82) f (Q σ QT , QσQT , Hi , QDQT ) = Qf (σ, σ, Hi , D)QT In the plastic constitutive equation formulated incorporating the yield and/or plastic potential function, the isotropy holds if the yield and/or plastic potential

6.5 Anisotropy

157

function is given by the function of stress invariants and scalar internal variables. Then, designating these functions by f , it must fulfill the equation.

f (QσQT , Hi) = Qf (σ, Hi)QT

(6.83)

In contrast, the anisotropic plastic constitutive equation is described by incorporating the yield and/or plastic potential function involving tensor-valued parameters in addition to the stress invariants and scalar internal variables. Then, Eqs. (6.82) and (6.83) do not hold in anisotropic constitutive equations.

6.5.2 Anisotropic Plastic Constitutive Equation If the monotonic loading proceeds towards a certain direction in the stress space, the hardening develops in that direction but the yield stress lowers in the opposite direction. This phenomenon is induced by the statically-indeterminable deformation of internal structure and is called the Bauschinger effect. To reflect this effect in the elastoplastic constitutive equation, the translation or the rotation of the yield surface is adopted widely. The translation of the yield surface, called the kinematic hardening, is realized by introducing the back stress developing towards the loading direction and replacing the stress tensor with the tensor given by subtracting the back stress tensor from the stress tensor. On the other hand, soils, which is the assembly of particles with weak adhesion between them, can bear a far larger compression stress than the tensile stress. Therefore, they exhibit a strong frictional property that the deviatoric yield stress increases with the pressure, while the yield surface only slightly includes the origin of the stress space. Therefore, once the yield surface translates leaving the origin, it can never come back to include the origin again because the yield surface contracts with the plastic volume expansion. Therefore, the kinematic hardening cannot be applied but the rotation of the yield surface, i.e. the rotational hardening, is pertinent to soils. Now, let the yield condition (6.30) be extended to describe the anisotropy by introducing the internal variables of second-order tensors as follows: f (σˆ , β ) = F ( H )

(6.84)

σˆ ≡ σ − α

(6.85)

where

α being the back stress (kinematic hardening variable) (Prager, 1956). Here, it is assumed that f in Eq. (6.84) is also the function of σ ˆ in the homogeneous degree-one fulfilling f ( sσˆ , β) = sf (σˆ , β) . Then, the yield surface (6.84) maintains the similar shape and orientation with respect to σ = α when β = const. β is the second-order tensor of non-dimension describing the rotation of the yield surface by

158

6 Basic Formulations for Elastoplastic Constitutive Equations

replacing the deviatoric stress ratio tensor σ' / p into

σ' / p − β and is called the

rotational hardening variable (Sekiguchi and Ohta, 1977). The isotropic deformation, i.e. expansion/contraction would not influence the anisotropy. Therefore, the anisotropic hardening variables α and β are assumed to be the deviatoric tensors, the evolution rules of which are given by D

αD = a(σ, Hi ) || D p || , β = b (σ, Hi ) || D p' ||

(6.86)

where a and b are functions of σ, H i and are deviatoric, fulfilling tra = trb = 0 . The symbol for deviatoric tensor is eliminated in (6.86)1 because of trD p = 0 in metals. The material-time derivative of Eq. (6.84) leads to the consistency condition

tr( ∂

ˆ β ˆβ f (σˆ , β) D σ) − tr(∂ f (σ, ) αD ) + tr( ∂ f (σ, ) βD ) = F ′ H• ∂β ∂σˆ ∂σˆ

(6.87)

Substituting Eqs. (6.37) and (6.86) into Eq. (6.87), one has

tr ( ∂

ˆ β ˆβ f (σˆ , β) D σ) − tr(∂ f (σ, ) a || D p|| ) + tr( ∂ f (σ, ) b || D p' || ) ∂β ∂σˆ ∂σˆ = F ′h (σ, Hi ; D p )

Further, substituting the associated flow rule ( λˆ factor)

ˆ ≡ ∂ f (σˆ , β) ˆ, N D p = λˆ N ∂σ

/ || ∂

(6.88)

： positive proportionality

f (σˆ , β) ∂σ ||

(6.89)

into Eq. (6.87) and noting ∂ f (σˆ , β) / ∂ σ = ∂ f (σˆ , β) / ∂ σˆ , it follows that

tr( ∂

ˆ ˆ f (σˆ , β) D σ) − tr(∂ f (σ, β) a λˆ) + tr(∂ f (σ, β)b || Nˆ ' || λˆ ) ∂β ∂ σˆ ∂ σˆ ˆ) = F ′λˆ h ( σ, H i ; N

(6.90)

Taking account of the Euler’s theorem for homogeneous function in order-one, i.e.

∂ f (σˆ , β) = ∂ σˆ

f (σˆ , β) tr ∂ σ ∂ σˆ ˆ = f (σˆ , β) N ˆ = F N ˆ N ˆ σˆ ) ˆ ˆ σˆ ) ˆ ) tr(N σ tr(N tr(N

(

)

(6.91)

6.6 Incorporation of Tangential-Inelastic Strain Rate

159

into Eq. (6.90), it is obtained that

F tr(N ˆ σD ) − F tr(N ˆ a) λˆ + tr( ∂ f (σˆ , β) b || N ˆ ' || ) λˆ ˆ ∂β ˆ σˆ ) tr(Nσˆ ) tr(N ˆ) = F ′λˆ h (σ, H i ; N

(6.92)

from which we obtains

λˆ =

=

D

ˆ σ) tr(N F ′ σ, H ; ˆ ˆ ' || ) tr(N ˆ σˆ ) ˆ a) − 1 ( ∂ f (σˆ , β) b || N ˆ σˆ ) + tr(N ( i N ) tr(N Fh F ∂β ˆ σD ) tr(N

(6.93)

ˆ ) − 1 (∂ f (σˆ , β) b || N ˆ ' ||)}σˆ + a] ˆ [{( F ′ h (σ, H i ; N tr N F F ∂β

)

(

Consequently, we have

λˆ =

ˆ σD ) tr(N Mˆ p

，D

p

=

ˆ σD ) tr(N ˆ N Mˆ p

(6.94)

where

ˆ ) − 1 ( ∂ f (σˆ , β) b || N' ||)}σˆ + a ] ˆ [{( F ′ h (σ, H i ; N Mˆ p ≡ tr N F ∂β F

)

(

(6.95)

and D

D = E−1 σ +

D

ˆ σ) tr(N ˆ N Mˆ p

(6.96)

from which it holds that

Λˆ =

ˆ D) t r( NE

ˆ ED) tr(N p tr( ˆ ˆ ˆ) M + NEN

(6.97)

ˆ N

ˆ

ˆ = (E − E ⊗ EN ) D EN σD = ED − ˆ p ˆ ˆ) ˆ ˆ) M + tr( NEN Mˆ p + t r( NEN

(6.98)

6.6 Incorporation of Tangential-Inelastic Strain Rate As presented in Eqs. (6.40), (6.48) and (6.94), the inelastic strain rate, i.e. the plastic strain rate in the traditional constitutive equation has the following limitations.

160

6 Basic Formulations for Elastoplastic Constitutive Equations

(i) It depends solely on the stress rate component normal to the yield surface, called the normal stress rate, but is independent of the component tangential to the yield surface, called the tangential stress rate, since it is derived merely based on the consistency condition. (ii) The direction is determined solely by the current state of stress and internal variables but it is independent of the stress rate. (iii) The principal directions of plastic strain rate tensor coincide with those of stress tensor, exhibiting the so-called coaxiality, in the case of isotropy in which the direction of plastic strain rate depends only on the direction of stress by the fact described in 1.11. On the other hand, it has been verified by experiments that an inelastic strain rate that influences deformation considerably is induced by the deviatoric tangential stress rate, and it is called the tangential inelastic strain rate, in the non-proportional loading process deviating from the proportional loading path normal to the yield surface. Here, the mean part of the tangential stress rate does not induce an inelastic strain rate, as Rudnicki and Rice (1975) verified based on the fissure model. The tangential inelastic strain rate is induced considerably in the plastic instability phenomena with the strain localization induced by the generation of the shear band and it influences the macroscopic deformation and strength characteristics even in the macroscopically proportional loading process. To remedy these insufficiencies, vairous models have been proposed to date as follows: 1) Intersection of plural yield surfaces: Various models assuming the intersection of plural yield surfaces have been proposed (Batdorf and Budiansky, 1949; Koiter, 1953; Bland, 1957; Mandel, 1965; Hill, 1966; Sewell, 1973, 1974). The Koiter’s (1953) model has been adopted by Sewell (1973, 1974), but it is indicated that the applicability of the model is limited to the inception of uniaxial loading. Models in this category cannot describe the latent hardening pertinently and are not readily applicable to general loading processes (cf. Christoffersen and Hutchinson (1979)). 2) Corner theory: The singularity of outward-normal of the yield surface is introduced by assuming the conical corner or vertex at the stress point on the yield surface. Therefore, the direction of plastic strain rate can take a wide range surrounded by the outward-normal of the yield surface (Christoffersen and Hutchinson, 1979; Ito, 1979; Gotoh, 1985; Goya and Ito, 1991; Petryk and Thermann, 1997). There exist the two kinds of models: One kind is based on the assumption of an imaginary infinitesimal vertex and the other subsumes a finite projecting cone. The evolution rule of the cone cannot be formulated and the reloading from the cone surface after partial unloading cannot be described pertinently by the latter models. It was described by Hecker (1972; Ikegami, 1979) that the yield surface projects towards the loading direction generally but the formation of the so-called vertex is doubtful. 3) Hypoplasticity: This term was first used by Dafalias (1986) in the analogy to the term hypoelasticity introduced by Truesdell (1955) described in 5.3. Models in

6.6 Incorporation of Tangential-Inelastic Strain Rate

161

this category are classified into models in which the direction of plastic strain D D rate depends on the direction of the stress rate σ /||σ || (Mroz, 1966; Dafalias and Popov, 1977; Hughes and Shakib, 1986; Wang et al., 1990; Hashiguchi, 1993a) and the models in which the direction of the plastic strain rate depends on the direction of stress rate D/ || D|| (Hill, 1959; Simo, 1987; Hashiguchi, 1997; Kuroda and Tvergaard, 2001). The singularity in the filed of direction of plastic strain rate is introduced algebraically into these models, although it is done geometrically in the models described in 1 ) and 2 ). However, the magnitude of the plastic strain rate is derived from the consistency condition. Therefore, the plastic strain rate diminishes when the stress rate is directed tangentially to the yield surface, as in the traditional constitutive equations without the vertex. The constitutive equations described in 1)–3) possess the following problems. i) A formulation of pertinent model which fulfills the consistency condition and is applicable to the general loading process is difficult. ii) The stress rate-strain rate relation becomes nonlinear. Therefore, the inverse expression cannot be derived, which renders deformation analysis as difficult. 4)

J 2 - deformation theory: Budiansky (1959) and later Rudnicki and Rice (1975) incorporated the tangential-inelastic strain rate into the constitutive equation (6.57) with the isotropic Mises yield condition as follows: •e σ + φ (σ e ){σD ' − t r ( σ' σD ') σ' } D = E −1 σD + 3 1 σ 2 F' σ e ' ||σ ' || ||σ ' ||

(6.99)

which can be rewritten as •e D = E−1 σD + 3 1 σ e σ' + φ (σ e ){σD ' − 2 / 3 σ• e 2 F' σ

σ'

2 / 3σ e

σ• e σ' + φ (σ e ) σD = E −1 σD + ( 3 1 − φ (σ e )) σ ' e 2 F'

} (6.100)

where the rate-linearity is retained. On the other hand, Hencky’s deformation theory (Hencky, 1924) is described as

ε = E −1σ + φ (σ e ) σ'

(6.101)

The material-time derivative of Eq. (6.101) leads to

εD = E −1 σD + φ ' (σ e ) σ• e σ' + φ (σ e )σD '

(6.102)

162

6 Basic Formulations for Elastoplastic Constitutive Equations

Comparing Eq. (6.100) with Eq. (6.102), choosing F (σ e ) so as to fulfill

1 F ' (ε e p(σ e )) = 3 2 φ (σ e )+ φ ' (σ e )σ e and regarding εD as D , it is known that the

(6.103)

J 2 - deformation theory (6.99) coin-

cides with Hencky’s deformation theory (6.101). In what follows, extend the J 2 - deformation theory for the general yield condition unlimited to the Mises yield condition (Hashiguchi, 1998, 2005; Hashiguchi and Tsutsumi, 2001,2003) First, assume that the strain rate is decomposed additively into elastic and plastic strain rates in Eq. (6.28) and further assume the tangential-inelastic strain Dt as

．

D = De + D p + Dt

(6.104)

D which is decomwhere Dt is induced by the deviatoric tangential stress rate σ ' D and the deviatoric tangential stress posed into the deviatoric normal stress rate σ n' D (Fig. 6.7): rate σ ' t

σD' = σD n' + σD 't

(6.105)

σD ' = σD − 13 t r σD

(6.106)

where

⎫ ⎪⎪ ⎬ σD 't ≡ Iˆ' σD = σD ' − σD 'n (tr(Nˆ σD 't ) = 0) ⎪⎪ ⎭

σD 'n ≡ nˆ' ⊗ nˆ ' σD = tr(nˆ ' σD )nˆ'

f ( ˆ , β) nˆ' ≡ (∂ σ )' ∂σ

Iˆ' ≡ I ' − nˆ ' ⊗ nˆ '

( ∂ f ∂(σσˆ , β) )'

=

ˆ' N (||nˆ' || =1) ˆ ' || || N

' ≡ 12 (δ ik δ jl + δ il δ jk ) − 13 δ ijδ kl − nˆ'ijnˆ'kl ) ( Iˆijkl

I' ≡ I − 1 I ⊗ I 3

' ≡ 1 (δ ikδ jl + δ ilδ jk ) − 1 δ ijδ kl ) ( Iijkl 2 3

(6.107)

(6.108)

(6.109) (6.110)

In these expressions, the fourth-order tensor I ' plays the role of transforming an arbitrary second-order tensor into its deviatoric second-order tensor. Therefore, it might be called the deviatoric transformation (or projection) tensor. Furthermore, the fourth-order tensor Iˆ' plays the role of transforming an arbitrary

6.6 Incorporation of Tangential-Inelastic Strain Rate

163

second-order tensor into its deviatoric second-order tensor tangential to the yield surface. For that reason, it might be called the deviatoric tangential (or projection) tensor. Hereinafter, the deviatoric tangential tensor is denoted as ( )'t , i.e. Tt' ≡ Iˆ' T for arbitrary second-order tensor T . nˆ '

'

n

' t'

00

σ ij'

Yield surface

Fig. 6.7 Normal and tangential stress rates in the deviatoric stress plane

Now, assume that the tangential inelastic strain rate Dt is related linearly to the tangential deviatoric stress rate σD 't . Dt = T σD t' 2G

(6.111)

where T is a function of stress and internal variables in general. Substituting Eqs. (6.29), (6.94), (6.111) into Eq. (6.104), the strain rate is given by ˆ σD ) D tr( N ˆ + T σD ' D = E −1 σ + N 2G t Mˆ p ˆ ˆ = (E −1 + N ⊗ N + 2TG Iˆ' ) σD Mˆ p

(6.112)

For derivation of the inverse expression of Eq. (6.112), let the elastic modulus tensor (5.28) in the Hooke’s type be adopted. Therefore, note that Λˆ is given by Eq. (6.97) itself since it holds that tr( NE σD 't ) = 2Gtr( NσD 't ) = 0 because of Eq. (6.107)2.

164

6 Basic Formulations for Elastoplastic Constitutive Equations

Noting (5.28), it holds from Eq. (6.112) that D

ˆ σ) tr( N D ˆ ' + T σD ' D' = 1 σ' + N 2G t 2G p ˆ M

(6.113)

from which one has the following relation, considering ˆ=N ˆ 't ≡ Iˆ' N ˆ ' − tr(nˆ' N ˆ ' ) nˆ ' = 0 . N

(6.114)

D't = 1 (1 + T ) σD 't 2G

where (6.115)

D't ≡ Iˆ' D = D' − tr(nˆ ' D' )nˆ'

Substituting further Eq. (6.114) into Eq. (6.111), the tangential inelastic strain rate is given by

T Dt = 1 + T D't

(6.116)

The stress rate is derived from Eqs. (6.29), (6.89), (6.97) and (6.116) as follows:

σD = ED −

ˆ ) EN t r ( NED ˆ − 2GT D't 1+ T ˆ ˆ) Mˆ p + t r ( NEN

(

ˆ ⊗ = E − EN

(6.117)

ˆ EN − 2+GT Iˆ ' D p ˆ ˆ ˆ M + t r ( NEN ) 1 T

)

(6.118)

Here, since there exists the relation

tr(

∂f (σ) ∂f (σ) D ∂f (σ) E(D − D p − Dt )} = dF EDe ) = tr{ σ ) = tr( ∂σ ∂σ ∂σ

(6.119)

it holds that ∂f (σ) tr{ EDt } = 0 ∂σ

(6.120)

Then, it is known that the tangential inelastic strain rate D t does not influence the hardening behavior. Furthermore, the loading criterion is given by Eq. (6.69) as it is, and the mathematical structure is rate-linear identically to the common elastoplastic constitutive equation, whilst it is called sometimes the linear comparison material since it is linear in the continuation of loading. Consequently, no difficulty is brought in the analysis of boundary value problems.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

165

The tangential inelastic strain rate has no loading criterion and is induced only if the deviatoric tangential stress rate is induced, falling within the framework of hypoelasticity in which the complete integrability condition does not hold and the time-integration depends on the loading path. However, the continuity condition is violated because it falls within the framework of the conventional plasticity assuming the interior of the yield surface to be an elastic domain. The tangential inelastic strain rate is induced suddenly when the stress reaches the yield surface. Therefore, the range of application is limited to the proportional loading process in which the tangential component of stress rate is far smaller than the normal component.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory Strictly speaking, the elastic strain rate has to be formulated based on the hyperelastic constitutive equation in which the elastic deformation is determined uniquely for a given stress and the elastic work done during the stress cycle is zero. On the other and, these basic requirements are not fulfilled exactly in the hypoelasticity as described in Chapter 5. In addition, the stress is calculated directly for a given elastic deformation in the hyperelastic constitutive equation. On the other hand, the hypoelastic constitutive equation must be integrated in time in order to calculate the stress. Therein, it is required to adopt an appropriate corotational stress rate in order to predict the stress pertinently. One of hyperelastic-plastic constitutive equations is shown below, which is based on the postulate of Eq. (6.20) (cf. Mandel, 1973, 1974; Simo, 1998; Belytschko et al., 2001). The hyperelastic constitutive equation is given by replacing F to F e in Eq. (5.9) for the unit initial density as 2 • ψ (E e) • S = ∂ e e Ee ∂E ∂E

(6.121)

where −1

S ≡ Fe τFe

−T

e e E e ≡ 1 (C − I ) = 1 (F eT F e − I ), C ≡ F eT F e 2 2

(6.122) (6.123)

S is interpreted to be the pull-back of the Kirchhoff stress τ to the intermediate configuration. Here, note that S∗ = Fe ∗−1τ ∗Fe ∗−T = (QFe ) −1 Qτ QT (QFe ) −T = Fe −1τFe −T = S leading to the frame-indifference of the hyperelastic constitutive relation.

166

6 Basic Formulations for Elastoplastic Constitutive Equations

One can formulate the following relation from Eq. (6.19). D = De + D p

(6.124)

where •e

pFe D ≡ F eT DF e, De ≡ E = F eTDe F e , D p ≡ F eTD

(6.125)

Here, note that D p is expressed as T D p = 1 (Ce Lp + Lp Ce ) 2

(6.126)

because of −1 −1 T D p = F eT 1 {F e Lp F e + (F e Lp F e )T }Fe = 1 ( F eT F eLp + Lp F eT F e) 2 2

The following equation is obtained from Eq. (6.121), adopting Eq. (6.124). •

el p S = Cel De = C ( D − D )

(6.127)

where Cel ≡

e ∂ 2 ψ (E ) e e ∂E ∂E

(6.128)

Now, we adopt the following yield condition for the sake of simplicity of explanation, while the extension to the anisotropy is not difficult. (6.129)

f (S) = F ( H )

while f is the homogeneous function of S in degree-one. The material-time differentiation of Eq. (6.129) leads to tr(

• ∂f (S) • S) = F' H ∂S

(6.130)

Noting the relation

∂f (S ) = ∂S

∂f (S ) S) f (S ) ∂S NS = N = F (H ) N tr( NS S) S tr(NS S ) S tr(N S S )

tr(

NS ≡

∂f (S) ∂S

/ ||

∂f (S) ∂S

||

(6.131)

(6.132)

the consistency condition is given from Eq. (6.130) as • • tr(N S S) = F' H tr(NS S) F

(6.133)

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

167

Here, let the plastic flow rule be given by

L p = λ r (S )

(6.134)

where r(S) stands for the direction of L p , which is function of S . The substitution of Eq. (6.134) into Eq. (6.133), one has • tr(N S S ) = F' λ h (r )tr( N S S ) F

(6.135)

•

•

where h (S) is related to H as H = λ h (S) . It holds from Eqs. (6.134) and (6.135) that •

•

tr(N S S)

λ=

tr(N S S)

p

r (S) , L = F F' h ( ) ' h ( S) tr(N S S) tr(N S S) S F F

(6.136)

p

D is given from Eqs. (6.126) and (6.136) by •

p

D =

(6.137)

tr(N S S )

1 (Cer + rT Ce ) 2 F' h ( ) F S tr(N S S)

D is given from Eqs. (6.124), (6.127) and (6.137) as el −1

D=C

•

S+

•

tr(N S S )

(6.138)

1 (Cer + rT Ce ) 2 F' h ( ) F S tr(N S S)

from which the proportionality factor in the flow rule in Eq. (6.134) is described in terms of D as

Λ=

tr(N S Cel D)

(6.139)

F' h ( S) tr( N S S) + tr{NS Cel (Cer + rT Ce ) / 2 } F

It is obtained from Eqs. (6.138) and (6.139) that •

[

el −

S= C

Cel 1 (Cer + rT Ce ) ⊗ tr(N S Cel ) 2 F' h ( r) tr(N S S) + tr{NS Cel (Cer + rT Ce ) / 2 } D F

]

(6.140)

On the other hand, Lie derivative of S due to F p defined in 4.5, is given by •

Lvp S = S −Lp S − SLpT

(6.141)

168

6 Basic Formulations for Elastoplastic Constitutive Equations

noting

L vpS = F p (F p −1SF p −T )• F pT • p −1

= F p (F •p

= (− F F

•

• −T

SF p −T + F p −1 S F p −T + F p −1S F p

p −1

•

)F pT

•

) S + S + S ( − F p F p −1 )T

The substitution of Eqs. (6.134) and (6.140) into Eq. (6.141) leads to

L vp S = C ep D

(6.142)

where

C

ep ≡

C

el

1 Cel 2 (Cer + rT Ce ) + (r S + S r )}⊗ tr(N S Cel ) { −

(6.143)

F' h ( ) tr(N S S ) + tr{NS Cel (Cer + rT Ce ) / 2} F S

It holds from Eqs. (3.21), (4.57) and (6.122) for the Truesdell rate of Kirchhaff stress that Δ

τ = L v τ = F (F −1τF −T )• F T = F eF p ( F p −1F e−1F e S F e T F e −T F p −T )• F p T F e T = F eF p ( F p −1S F p − T )•F p T F e T = F e (Lvp S ) F eT

(6.144)

Substituting Eqs. (6.125) and (6.142), it is derived that Δ

τ = F e C ep DF e T = F e C ep ( F eT D F e) F e T = C ep D

(6.145)

⎛ Δ ij ⎞ ep e e ⎜τ = Fim (C mnpq Dpq ) Fjn ⎟ ⎜ = F e (C ep F e D F e ) F e = F e F e F e F eC ep D ⎟ im jn im jn kp lq mn pq kl ⎠ mn pq kp kl lq ⎝ where ep

ep C ijkl = Fime Fjen Fpke Fqel Cmn pq

(6.146)

Noting Eq. (4.57), Eq. (6.145) is rewritten as :

Δ

τ (= τ + Dτ + τD) = (C ep + Σ) D

(6.147)

by the Jaumann rate of Kirchhoff stress or, noting Eq. (4.54), :

: σ ( = τ − σ trD) = ( 1 C ep + Σ − σ ⊗ I )D J

J

by the Jaumann rate of Cauchy stress, where Σ is defined in Eq. (5.15).

(6.148)

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

169

In the above-mentioned constitutive equation, the flow rule (6.134) for L p has to be formulated rigorously so as to provide the plastic strain rate pertinently noting the associated flow rule. Further, it has to be done also for the spin due to the plastic deformation, i.e. W p = λ (r(S))a . Further discussions for the finite elastoplastic Experiment Elastic state

Prediction by convention al plasticity

Elastoplastic state St ress

St ress Unrealistic pred iction: Excessively high peak stress

Almost real prediction

0 Hardening

St rain

0

Strain

Softening

Fig. 6.8 Prediction of monotonic loading behavior by conventional plasticity Experiment Elastic state Stress

Elas to plastic state

Pr ediction by conven tional plasticity

Cyclic loading with constant st ress amplitud amplitude

0

Fig. 6.9 Cyclic loading behavior: Inability of conventional plasticity

Strain

170

6 Basic Formulations for Elastoplastic Constitutive Equations

strain theory are referred to the literatures (cf. Kleiber and Raniecki,1985; Raniecki and Mroz, 1990; Raniecki and Nguyen, 2005; Raniecki et al., 2008). The conventional elastoplasticity described in this chapter is premised on the assumption that the interior of yield surface is a purely elastic domain. Therefore, the relation of stress rate vs. strain rate is predicted to change abruptly at the moment when the stress reaches the yield surface. Therefore, the smooth stress-strain curve observed in real materials is not predicted as shown in Fig. 6.8. This results in the serious defect in the prediction of softening behavior. Further, only an elastic deformation is repeated for the cyclic loading of stress below the yield stress. In fact, however, plastic deformation is accumulated for stress cycles less than the yield stress and the strain is amplified leading to the failure as depicted in Fig. 6.9. Therefore, it has various limitations in the application to the mechanical design of machines and structures in engineering practice.

Chapter 7

Unconventional Elastoplasticity Model: Subloading Surface Model 7 Unconventional Model: Subloading SurfaceElastoplasticity Model

Elastoplastic constitutive equations with the yield surface enclosing the elastic domain possess many limitations in the description of elastoplastic deformation, as explained in the last chapter. They are designated as the conventional model in Drucker’s (1988) classification of plasticity models. Various unconventional elastoplasticity models have been proposed, which are intended to describe the plastic strain rate induced by the rate of stress inside the yield surface. Among them, the subloading surface model is the only pertinent model fulfilling the mechanical requirements for elastoplastic constitutive equations. These mechanical requirements are first described and then the subloading surface model is explained in detail.

7.1 Mechanical Requirements There exist various mechanical requirements, e.g., the thermodynamic restriction and the objectivity for constitutive equations. Among them, the continuity and the smoothness conditions are violated in many elastoplasticity models, although their importance for formulation of constitutive equations has not been sufficiently recognized to date. Before formulation of the plastic strain rate, these requirements will be explained below (Hashiguchi, 1993a, b, 1997, 2000).

7.1.1 Continuity Condition It is observed in experiments that “the stress rate changes continuously for a continuous change of the strain rate”. This fact is called the continuity condition and is expressed mathematically as follows (Hashiguchi, 1993a, b, 1997; 2000). D D lim σ (σ, Hi ; D + δ D) = σ (σ, Hi ; D)

δ D→ 0

(7.1)

where H i (i =1, 2, 3, • ••) collectively denotes scalar-valued or tensor-valued internal state variables. In addition, δ ( ) stands for an infinitesimal variation. The response of the stress rate to the input of strain rate in the current state of stress and D internal variables is designated by σ (σ, Hi ; D). Uniqueness of the solution is not guaranteed in constitutive equations violating the continuity condition, predicting

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 171–189. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

172

7 Unconventional Elastoplasticity Model: Subloading Surface Model

jump

jump

Input : D

D Out put : ı

Fig. 7.1 Violation of continuity condition

different stresses or strains. The violation of this condition is schematically shown in Fig. 7.1. Ordinary elastoplastic constitutive equations, in which the plastic strain rate is derived obeying the consistency condition, fulfill the continuity condition. As described later, however, no elastoplastic constitutive equation fulfills it except for the subloading surface model when they are extended to describe the tangential inelastic strain rate. The concept of the continuity condition was first advocated by Prager (1949). However, a mathematical expression of this condition was not given. The condition was defined as the continuity of strain rate to the input of stress rate by Prager (1949) inversely to the definition given above. However, the identical stress rate directed the inward of yield surface can induce different strain rates in loading and unloading states in a softening material. Here, it is noteworthy that a stress rate cannot be given arbitrarily since there exists a limitation in strength of materials although a strain rate can be given arbitrarily. For that reason, the Prager’s (1949) notion does not hold in the general loading state including softening and the perfectly plastic states (Fig. 7.2).

7.1.2 Smoothness Condition It is observed in experiments that “the stress rate induced by the identical strain rate changes continuously for a continuous change of stress state”. This fact is called the smoothness condition and is expressed mathematically as follows: D lim σ (σ + δ σ, H i ; D ) = σD (σ, H i ; D )

δ σ →0

(7.2)

The rate-linear constitutive equation is described as

σD = M ep (σ, H i) D

(7.3)

7.1 Mechanical Requirements

173

σ

Input: d σ (<0)

unloading

0

loading

Output: d ε d ε (<0) (>0)

ε

Tangent plane of yield surface

Output : D D Input : ı Fig. 7.2 Impertinence of Prager’s (1949) continuity condition (illustrated for softening state)

where the fourth-order tensor Mep is the elastoplastic modulus, which is a function of the stress and internal variables, can be described generally as D

M ep = ∂ σ . ∂D

(7.4)

174

7 Unconventional Elastoplasticity Model: Subloading Surface Model

Consequently, Eq. (7.2) can be rewritten as

lim M ep (σ + δ σ, H i ) = M ep (σ, H i )

δ σ →0

(7.5)

Constitutive equations violating the smoothness condition cannot predict a smooth stress-strain curve. Therefore, they cannot describe softening behavior pertinently, as depicted in Fig. 6.8. Further, it cannot predict the strain accumulation for cyclic loading of stress amplitude less than the yield stress, as depicted in Fig. 6.9. Among the existing constitutive models only the subloading surface model always fulfills the smoothness condition.

7.2 Subloading Surface Model The basic concept and equations for the subloading surface model (Hashiguchi and Ueno, 1977; Hashiguchi, 1978, 1980, 1989) are described below. This is the only model fulfilling the mathematical requirements described in 7.1. For the formulation of plastic strain rate induced by the rate of stress inside the yield surface let the following assumptions be incorporated for the loading process (D p ≠ 0) , which would be quite natural properties in plastic deformation. a)

b) c) d) e)

Only an elastic strain rate is induced up to a certain stress level. Therefore, the stress changes at infinite rate up to that stress level even if the plastic strain rate is infinitesimal. The stress always approaches the yield surface. The ratio of the magnitude of stress rate to that of plastic strain rate decreases as the stress approaches the yield surface. The stress changes along the yield surface in the plastic loading process when it reaches the yield surface. It does not go out from the yield surface. A conventional elastoplastic constitutive equation holds when the stress lies on the yield surface.

To formulate an unconventional elastoplastic constitutive equation based on these assumptions, it is necessary to adopt an appropriate measure describing the degree of approach to the yield state. Then, let the subloading surface, be introduced, which always passes through the current stress point and has similar shape and orientation to the yield surface, while the yield surface is renamed the normal-yield surface. Here, the similar shape and orientation of surfaces imply the following geometrical properties. i ) All lines connecting an arbitrary point on or inside the subloading surface and it conjugate point on or within the normal-yield surface join at the similarity-center. ii ) All ratios of length of an arbitrary line-element connecting two points on or inside the subloading surface and that of the conjugate line-element connecting two conjugate points on or inside the normal-yield surface are identical. The ratio is called the similarity-ratio which coincides with the ratio of the sizes of these surfaces.

7.1 Mechanical Requirements

175

Consequently, the subloading surface (see Fig. 7.3) can be described as

f (σ ) = RF ( H )

(7.6)

where, R (0 ≤ R ≤ 1) is the ratio of the size of the subloading surface to that of normal-yield surface, called as the normal-yield ratio. It plays the role of description for the approaching degree of stress to the normal-yield surface. The state R = 0 ( f = 0 ) corresponds to the null stress state in which a purely elastic deformation behavior occurs, the state 0 < R < 1 ( 0 < f < F ) to the sub-yield state, and the state R = 1 ( f = F ) to the normal-yield state in which the stress lies on the normal-yield surface for which the plastic strain rate has been formulated in the conventional plasticity. It should be noted that the subloading surface is not assumed independently from the normal-yield surface but is merely the shadow surface projected from the normal-yield surface. Therefore, the subloading surface coincides completely with the normal-yield surface when they contact mutually, leading to R = 1 . Eventually, there exists only one independent surface, i.e. the normal-yield surface in the subloading surface model identically to the conventional plasticity. On the other hand, plural independent yield (loading) surfaces are assumed and they contact mutually at one point, resulting in the discontinuity of plastic modulus, in unconventional models, e.g. the multi surface model (Mroz, 1967; Iwan, 1967) and the two surface models (Dafalias, 1975; Krieg, 1975) and the infinite surface model (Mroz et al., 1981) delineated in the next chapter. Normal- yield surface Subloading surface

f ( ı ) = F (H )

f ( ı ) = R F (H )

Elastic region

ı

Re

R

0

Fig. 7.3 Normal-yield and subloading surfaces

1

σij

176

7 Unconventional Elastoplasticity Model: Subloading Surface Model

Regarding the plastic strain rate formulation, it is noteworthy that the following equation should be fulfilled to incorporate assumptions a) and e) described above, noting Eqs. (6.40) and (6.41). ⎧0 for R = 0 ⎪ ⎪ tr(N σD ) Dp = ⎨ N for R =1 ⎪ F' h (σ, H i ; N) tr(Nσ ) ⎪F ⎩

(7.7)

The simplest equation fulfilling Eq. (7.7) would be given by the following equation if adopting the interpolation method.

D p = Rn

tr(N σD ) F' h (σ, H ; N) tr(Nσ) i F

N

(7.8)

However, Eq. (7.8) cannot be adopted for materials exhibiting softening behavior induced by the plastic volumetric strain as seen in soils in which when the stress reaches the critical state line, it holds that h (σ, H i ; N ) = t rN = 0 , and thus the stress cannot go up that line. In stead, we can assume various equations other than Eq. (7.8) fulfilling Eq. (7.7), e.g. Dp =

tr(N σD ) F' h(σ, H ; N) tr(Nσ ) + ( R ) S i F

N

(7.9)

and

Dp =

tr(N σD )

{ FF' h(σ, H i ; N) + S ( R)}tr(Nσ)

N

(7.10)

where S ( R) is the monotonically-decreasing function of R fulfilling

⎧→ +∞ for R → 0 S ( R) ⎨ for R = 1 ⎩= 0

(7.11)

Here, it remains ambiguous which equation among them is rigorous. In what follows, let a pertinent equation for plastic strain rate be formulated by incorporating the consistency condition for the subloading surface. The material-time derivative of Eq. (7.6) is given as tr ( ∂

f (σ) σD ) = RF' H• + R• F ∂σ

(7.12)

7.1 Mechanical Requirements

177

Equation (7.12) itself cannot play the role of the consistency condition for the • derivation of plastic strain rate because it includes the rate variable R , which is not • related to the plastic strain rate at this stage, although H is related to the plastic strain rate in Eq. (6.37). To embody Eq. (7.12) as the consistency condition, let the • evolution rule of R , i.e. R be formulated to fulfill the following conditions in the p loading process (D ≠ 0) , based on the assumptions a )-e ) described in the foregoing. a)

The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate •

is infinite up to a certain value of normal-yield ratio: R / ||D p || → ∞ for R ≤ R e , where Re (<1) is the material constant describing the elastic limit of R (Tsutsumi et al, 2006), whereas Re ≥ 0.2 for many metals but it can be put that b)

c)

d)

e) f)

Re = 0 for soils. The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate is positive before the subloading surface coincides with the normal-yield sur• face: R / ||D p || > 0 for 0 < R < 1 . The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate •

p

decreases monotonically as the normal-yield ratio increases: R / ||D || is the monotonically-decreasing function of R . The subloading surface does not expand over the normal-yield surface. Then, the ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate becomes zero when the subloading surface coincides with the normal-yield • surface R / ||D p || = 0 for R =1 . A conventional elastoplastic constitutive equation holds as it is in the normal-yield state: Eq. (6.40) with Eq. (6.41) holds for R = 1 . The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate becomes negative if the subloading surface becomes larger than the nor•

mal-yield surface: R / ||D p || < 0 for R>1 . Based on these assumptions, let the following evolution equation of the normal-yield ratio R be assumed. •

p R = U ( R)||D p || for D ≠ 0

(7.13)

where U is a monotonically decreasing function of R , fulfilling the following conditions (see Fig. 7.4).

⎧→ +∞ for 0 ≤ R ≤ R e (quasi - elastic state) ⎪> 0 for R < R < 1 (sub - yield state), e ⎪ U ( R) ⎨ ⎪= 0 for R = 1 (normal - yield state), ⎪⎩< 0 for R > 1 (over normal - yield state)

(7.14)

178

7 Unconventional Elastoplasticity Model: Subloading Surface Model

The stress-controlling ability attracting stress to normal-yield surface is furnished in this model incorporating Eq. (7.14). Let the function U satisfying Eq. (7.14) be simply given by

〈 R − Re〉 U ( R ) = u cot( π2 1 − R ) e

(7.15)

where u is a material constant and 〈 〉 is the McCauley’s bracket defined by 〈 s〉 = ( s + | s |)/2 , i.e. 〈 s〉 = s for s ≥ 0 and 〈 s〉 = 0 for s<0 for an arbitrary scalar variable. Equation (7.13) with Eq. (7.15) can be integrated analytically as p ⎫ R0 − Re π exp u ε − ε 0p π )} + R ⎪ ) − e ( 1 − Re 2 1 − Re 2 ⎪ ⎪ (7.16) − R R e π 0 cos( ) ⎬ − 1 2 R e ε p − ε 0p = (1 − Re ) π2 ln ⎪ R R ⎪ cos ( −− e π ) 1 Re 2 ⎪ ⎭

R = π2 (1 − Re )cos −1{cos(

under the initial condition ε p = ε 0p : R = R 0 , where ε p ≡ ∫||D p|| dt (t: time) . The following equation for U has been used after Hashiguchi (1989) but the analytical integration cannot be obtained.

− 〉 U ( R) = −u ln 〈 R Re 1 − Re •

R = U (R ) || D p ||

0

Re

1

R

Fig. 7.4 Function U ( R ) in the evolution rule of normal-yield ratio

(7.17)

7.1 Mechanical Requirements

179

The value of R is calculated by substituting the current values of stress and internal variables into the equation of the subloading surface. It can also be calculated by integrating the evolution equation of R , i.e. Eq. (7.13) in the loading process ( D p ≠ 0 ) . However, it must be calculated directly from the equation of the subloading surface in the unloading process because the evolution equation, Eq. (7.13), does not hold in that process. The substitution of Eq. (6.37) and (7.13) into Eq. (7.12) yields the consistency condition extended for the subloading surface:

tr( ∂

f (σ ) D σ) = RF ′ h (σ, H i ; D p ) + U ( R) ||D p|| F ∂σ

(7.18)

Here, note that Eq. (7.18) reduces to the consistency condition (6.43) in the conventional plasticity for R = 1 . Therefore, assumption e) in the foregoing was also fulfilled. Further, substituting the associated flow rule

Dp = λ N

(7.19)

into Eq. (7.18) and using the relation

∂ f (σ ) = ∂σ

σ t r ∂f ( ) σ f (σ ) ∂σ N= N = RF N tr(N σ ) tr(N σ ) tr(N σ )

(

)

(7.20)

based on the Euler’s theorem of the homogeneous function in degree-one, it follows that

RF tr (N σD ) = RF ′ λ h(σ, H i ; N) + U ( R ) λ F tr(Nσ) from which one has

λ=

D

tr(N σ) p M

(7.21)

Consequently, p

D

D =

tr(N σ) p N M

(7.22)

where

M p≡(

F ′ h (σ, H ; N) + U ) ( Nσ) i R tr F

(7.23)

Substituting Eq. (6.29) and (7.22) into Eq. (6.28), the strain rate is given by D

D = E −1 σ +

D

tr(N σ ) N Mp

(7.24)

180

7 Unconventional Elastoplasticity Model: Subloading Surface Model

from which we have

Λ=

tr( NED) M + tr( NEN)

(7.25)

p

The stress rate is described from Eqs. (7.24) and (7.25) as

σD = ED − E 〈

tr(NED) 〉N M p + tr( NEN)

(7.26)

The plastic strain rate is induced depending on the direction of strain rate, even in the subyield surface. Therefore, a judgment of whether or not the yield condition is satisfied is not required. Therefore, the loading criterion (6.68) is simplified as Dp ≠ 0 : Λ > 0

⎫⎪ ⎬ D = 0 : otherwise ⎪⎭

(7.27)

p

The following relation is obtained from Eq. (7.26) in the loading state (D p ≠ 0) . D

tr( N σ) =

Mp tr(N ED) M + tr( NEN)

(7.28)

p

It is known from Eq. (7.28) that the subloading surface expands or contracts dep pending on the sign of M . Then, let it be defined that

⎫ ⎪ ⎪⎪ D M p = 0 → tr( N σ) = 0 : subloading nonhardening ⎬ ⎪ ⎪ M p < 0 → tr( N σD ) < 0 : subloading softening ⎪⎭ M p > 0 → tr( N σD ) > 0 : subloading hardening

(7.29)

On the other hand, the normal-yield surface expands or contracts depending on the sign of the rate of isotropic hardening function F• . Then, let it be defined alternatively that

⎫ ⎪ ⎪⎪ • F = 0 : normal - isotropic nonhardening ⎬ ⎪ • ⎪ F < 0 : normal - isotropic softening ⎪⎭ •

F > 0 : normal - isotropic hardening

(7.30)

On the other hand, a similar equation to Eq. (7.28) holds also for the conven• tional plasticity. Nevertheless, the signs of t r( N σD ) and F are the same because of

7.3 Salient Features of Subloading Surface Model

181

Eq. (6.35) of the consistency condition. Further, the positiveness of λ in Eq. (6.40) engenders the coincidence of the signs of t r( N σD ) and M p . Therefore, the sings of •

tr( N σD ) , M p and F coincide mutually. Eventually, the distinction between Eqs. (7.29) and (7.30) does not exist in the conventional plasticity.

7.3 Salient Features of Subloading Surface Model The subloading surface model reduces to the conventional plasticity model with the yield surface enclosing the purely elastic domain by putting u → ∞ or Re = 1. It is improved substantially from the conventional model and has noticeable advantages compared even with the other unconventional model, as described below. It predicts a smooth response in a smooth monotonic loading process, i.e. a smooth relation of axial stress and axial logarithmic strain in uniaxial monotonic loading for example, as shown in Fig. 7.5.

Conventional plasticity model (u → ∞)

σ1

ic

σ1

Ela st

i)

Subloading surface model

ı Re

0

R 1

ε1

0

σ2

Subloading surface

σ3

Normal-yield surface Fig. 7.5 Smooth stress-strain curve predicted by the subloading surface

The influence of the material constant u in the evolution rule of the normal-yield ratio R on the curvature of stress-strain curve is depicted in Fig. 7.6. The stress-strain curve for u → ∞ coincides with that predicted by the conventional constitutive equation. For the smaller value of u , a more gentle

182

7 Unconventional Elastoplasticity Model: Subloading Surface Model

transition from the elastic to plastic state, called the elastic-plastic transition, is predicted. The decrease in the value of Re also engenders the prediction of smoother elastic-plastic transition. The smoothness condition is fulfilled always even for R e ≠ 0 , because the ratio of the rate of normal-yield ratio to the • magnitude of plastic strain rate, i.e. R / ||D p|| , changes continuously from the infinite value at R = Re to zero in the normal-yield state ( R = 1). On the other hand, a non-smooth response is predicted by the conventional constitutive model, violating the smoothness condition at the yield point. The nonsmooth response is predicted also by the unconventional plasticity models other than the subloading surface model, e.g. the multi surface model (Mroz, 1967; Iwan, 1967), the two surface model (Dafalias, 1975; Krieg, 1975) and the nonlinear kinematic hardening model (Armstrong and Fredericson, 1966; Chaboche et al., 1979) because the small yield surface enclosing the purely-elastic domain and/or plural subuield surfaces with different sizes are assumed and thus the smoothness condition is violated when the stress reaches theses surfaces, exhibiting the discontinuous mechanical response. ii) The stress always lies on the subloading surface which plays the role of the loading surface. Therefore, only the judgment for the sign of the proportionality factor Λ is required in the loading criterion for the subloading surface model. On the other hand, for the plasticity model assuming the yield surface enclosing the purely-elastic domain, a judgment whether or not the stress lies on the yield surface is required in addition to a judgment for the sign of Λ . Moreover, a judgment on which loading surface the current stress lies is required in the unconventional models assuming the subyield surface(s). iii) A stress is automatically drawn back to the normal-yield surface even if it goes • out from that surface because it is formulated that R < 0 for R > 1 (over normal-yield state) in Eq. (7.13) with condition (7.14) (see Fig. 7.7). A stable and robust calculation can be executed even by rough loading steps as will be verified quantitatively in Chapter 14. On the other hand, the numerical calculation by the conventional plasticity model requires the algorithm to pull back the stress to the yield surface. Unconventional models other than the subloading surface model also requires the operation to pull back the stress to a current loading surface, as described in the next chapter. Eventually, the subloading surface model can describe pertinently the monotonic loading process. Besides, it is necessary to describe rigorously the deformation behavior of materials undergoing softening behavior with the plastic volumetric strain, e.g. soils as will be delineated in Chapter 11. In addition, the subloading surface model possesses the high ability in numarical calculation, while this distinguishable advantage will be quantitatively verified in Chapter 14.

7.3 Salient Features of Subloading Surface Model

s t ic na l p la io t n e v C on

σ

u =п

di n loa b u S

u

c de

a re

se

gs

it y mo

183

de l

o ce m a f r u

del

s

ε

0

Fig. 7.6 Variation in curvature of stress-strain curve due to material constant rule of normal-yield ratio

u

in evolution

•

R > 1: R < 0 σ

• R < 1: R > 0

R=

0 • p p R = U (R ) || D || for D ≠ 0

1:

R=• 0

εp

→ +∞ for 0 ≤ R ≤ Re (almost elastic state) ° > 0 for R < R< 1 (sub - yield state), e ° U (R ) ® ° = 0 for R = 1 (normal - yield state), °¯ < 0 for R > 1 (over normal - yield state) Fig. 7.7 Stress-controlling function in subloading surface model: Stress is automatically attracted to yield surface in the plastic loading process

184

7 Unconventional Elastoplasticity Model: Subloading Surface Model

7.4 On Bounding Surface and Bounding Surface Model The terms bounding surface and bounding surface model are widely used for models falling within the framework of unconventional plasticity describing the plastic strain rate induced by the rate of stress inside the yield surface. This nomenclature was named by Y. F. Dafalias (1975), who also coined the terms plastic spin (Dafalias, 1985) and hypoplasticity (Dafalias, 1986). The only concrete model proposed by Dafalias as the bounding surface model is the two-surface model (Dafalias and Popov, 1975), in which a small subyield surface is assumed inside the yield surface, which encloses the purely elastic domain and translates maintaining the constant ratio of the size to the size of yield surface, and the single surface model (Dafalias and Popov, 1977), in which the subyield surface shrinks to a point. However, note the following facts. 1) The so-called bounding surface is no more than the yield surface that has been used historically in the field of plasticity. The term yield surface has a clear physical meaning that the plastic deformation begins when stress reaches it; it also has the geometrical meaning that the stress cannot go out from it in the quasi-static deformation process. In contrast, the phrase bounding surface has only a geometrical meaning but has no physical meaning. 2) The yield surface always exists. However, the stress goes over the yield surface in the deformation process at a high rate as represented by the overstress model in the viscoplastic constitutive models. Therefore, no surface exists which bounds the stress except for the quasi-static deformation process. Consequently, the phrase bounding surface has no generality. 3) The bounding surface is to be the yield surface itself. Therefore, the term bounding surface model induces the confusion that all unconventional plasticity models inheriting the yield surface belong to the bounding surface model. Words expressing salient features of each model should be used for names of models to avoid confusion. In fact, researchers aside from Dafalias, e.g. Krieg (1975) uses the term limit surface to describe the novel loading surface in his two surface model, Mroz (1967) uses outmost surface in his multi surface model, and Hashiguchi (1989) uses normal-yield surface in his subloading surface model instead of yield surface. However, they use these words only in a limited sense for naming elements in their models: they never use these words as names of their proposed models such as the limit surface model, the outmost surface model, or the normal-yield surface model. Furthermore, Dafalias uses the phrase bounding surface model with a radial mapping (Dafalias and Herrmann, 1980). Nevertheless, the model has a physical and mathematical structure that differs from that of a two-surface model but has similar structure to that of the subloading surface model proposed in 1977 three years earlier than 1980 when Dafalias began to write the articles on the bounding surface model with a radial mapping. Furthermore, note that it involves various immature and impertinent formulations as described below.

7.4 On Bounding Surface and Bounding Surface Model

185

In the bounding surface model with a radial mapping, any surface other than the yield surface is adopted, but the following ratio is adopted as the measure to describe the degree of approaching the yield surface. b ≡ ||σ y || / ||σ ||

(7.31)

which is the ratio of the magnitude of current stress σ to the magnitude of conjugate stress σ y on the yield surface. Then, the plastic modulus M p in the plastic strain rate of the conventional plasticity, i.e. Eq. (6.41) is modified as

⎛ ⎧→ ∞ for b ≥ be ⎞ − M p → M p + Hˆ 〈 b 1 〉 ⎜ = ⎨ p be − b ⎜ M for b = 1 ⎟⎟ ⎝ ⎩ ⎠

(7.32)

where Hˆ is the function of stress and internal variables and b e is the value of the variable b at the elastic limit. Here, note the following facts. i. The variable b (1 ≤ b ≤ ∞) is merely the inverse of normal-yield ratio R (0 ≤ R ≤ 1) in the subloading surface model. ii. The plastic modulus M p in the bounding surface model with a radial mapping is given by the interpolation rule between the null stress and the conjugate stress on the yield surface, where no consistency condition is introduced as confirmed from the statement “No consistency condition f• = 0 is required for stress points inside F = 0 , since now f = 0 is always defined at any σ ij .” (P. 978: Dafalias, 1986)), whereas the consistency condition for the subloading surface is introduced into the subloading surface model. Various equations other than Eq. (7.32) can be assumed for the plastic modulus if an easy-going interpolation method is adopted. It might be readily apparent that Eq. (7.32) differs substantially from the plastic modulus of Eq. (7.23) which is derived rigorously from the consistency condition obtained by incorporating the assumption that the normal-yield ratio approaches unity in the plastic loading process. iii. Therefore, it is not guaranteed that the stress approaches the yield surface in the plastic loading process. On the other hand, the subloading surface model possesses a stress controlling function to attract the stress to the yield surface in the plastic loading process even if the stress goes out from the yield surface in the numerical calculation by the finite stress increments. iv. A formulation for describing cyclic loading behavior has not been given for the bounding surface model with a radial mapping. On the other hand, it has been attained in the subloading surface model as the extended subloading surface model (Hashiguchi, 1989) by making the similarity-center of the normal-yield and the subloading surfaces move with the plastic strain rate as described in detail in the next chapter.

186

7 Unconventional Elastoplasticity Model: Subloading Surface Model

Eventually, it can be concluded for the bounding surface model with radial mapping as follows: I) The bounding surface is substantially the synonym of the yield surface although it does not express any physical meaning. Therefore, the bounding surface model would cause confusion as if all models adopting the yield surface belong to this model, while in fact it is insisted by Dafalias that “the model with “stress reversal surfaces” (infinite surface model of Mroz et al., 1981) proposed for soils can be classified as a radial mapping model” (p. 981: Dafalias, 1986) in addition to the impertinent assessment on the subloading surface model. It is desirable to make effort for concrete formulation of pertinent model rather than only coining new terms. For that reason, the words yield surface should not be replaced with the term bounding surface which would not have to be used for the steady and sound development of elastoplasticity. II) The bounding surface model with radial mapping falls within the framework of the subloading surface model but it is not formulated rationally, whereas the rigorous formulations including the description of cyclic loading behavior have been given by the subloading surface model.

7.5 Incorporation of Anisotropy A subloading surface based on the yield surface in Eq. (6.84) with the kinematic and rotational hardening is described as (7.33)

f (σˆ , β ) = RF ( H )

The material-time derivative of Eq. (7.33) is given as

ˆ , β) Dβ • f (σˆ , β) σD f (σˆ , β) α D + t r ∂ f (σ • − tr ∂ tr ∂ = R F + RF ′ H β ∂ ˆ ˆ ∂σ ∂σ

(

) (

) (

)

(7.34)

Substituting Eq. (6.37), (6.86), (7.13) and the associated flow rule ˆ D p = λ N

(7.35)

(7.34) is rewritten as

tr(∂

ˆ ˆ f (σˆ , β) D σ) − tr(∂ f (σ, β) a λ ) + tr( ∂ f (σ, β) b λ|| Nˆ ' ||) ∂β ∂ σˆ ∂ σˆ ˆ ) + U λ F = RF ′λ h ( σ, H i ; N

(7.36)

Noting ∂ f (σˆ , β) ∂ σˆ =

f (σˆ , β) tr ∂ σ ∂ σˆ ˆ = f (σˆ , β) N ˆ = RF N ˆ N ˆ ˆ σˆ ) ˆ σˆ ) ˆ tr(N σ) tr(N tr(N

(

)

(7.37)

7.6 Incorporation of Tangential Inelastic Strain Rate

187

Eq. (7.36) is rewritten as

ˆ ˆ σD ) − tr(N ˆ a λ ) + tr(N σˆ ) t r ( ∂ f (σˆ , β) b λ || N ˆ ' ||) tr(N RF ∂β ′ ˆ ) + U λ ) tr(N ˆ σˆ ) = ( F λ h ( σ, H i ; N R F

(7.38)

from which one has

λ =

ˆ σD ) ˆ σD ) tr(N tr(N ˆ , Dp = N p M M p

(7.39)

where M

p

ˆ ' || ∂ f (σˆ , β) ˆ ) + U − || N ˆ [{ F ′ h (σ, H i ; N ≡ tr N ( ∂ β b)} σˆ + a ] R RF F

[

]

(7.40)

7.6 Incorporation of Tangential Inelastic Strain Rate Let the tangential inelastic strain rate be incorporated into the above-mentioned subloading surface model in the following (Hashiguchi, 1998, 2005; Hashiguchi and Tsutsumi, 2003; Hashiguchi and Protasov, 2004; Khojastepour and Hashiguchi, 2004a, b). Assume that T in Eq. (6.111) in 6.5 is a monotonically-increasing function of R in addition to the stress and the internal variable, i.e.

T ≡ ξ Rτ

(7.41)

where τ is the material constant and ξ is the function of the stress σ and the internal variables Hi as ξ = ξ (σ, Hi )

(7.42)

Adding Eq. (6.111) with Eq. (7.41) into Eq. (7.24) yields D

D = E −1 σ +

τ ˆ σD ) tr(N ˆ + ξ R σD 't N p 2G M

(7.43)

or D

D = E −1 σ +

τ ˆ σD ) tr(N ˆ + ξ R Iˆ' σD N p 2G M

ˆ ˆ ξ Rτ D D = (E −1 + N ⊗pN + 2G Iˆ' ) σ M

(7.44) (7.45)

188

7 Unconventional Elastoplasticity Model: Subloading Surface Model

from which the stress rate is given as σD = ED − 〈

τ ˆ ) t r ( NED ˆ − 2G ξ R D't EN 〉 τ ˆ ˆ) 1+ ξ R M p + t r ( NEN

(7.46)

or ξ Rτ

− 2G Iˆ' D σD = E − EN ⊗ p EN M + tr (NEN) 1 + ξ Rτ

(

)

(7.47)

The deviatoric normal and tangential stress rates for the subloading surface model are shown in Fig. 7.8. This model fulfills the continuity and the smoothness condition as shown in Fig. 7.9. On the other hand, all models other than the subloading surface model violate the smoothness condition. Therefore, they also violate the continuity condition as illustrated for the J2 -deformation model of Rudnicki and Rice (1975) in Fig. 7.9. The subloading surface model has been applied to metals (Hashiguchi, 1980; 1989; Hashiguchi and Yoshimaru, 1995; Hashiguchi and Tsutsumi, 2001; Hashiguchi and Protasov, 2004; Khojastehpor et al., 2006; Tsutsumi et al., 2005) and soils (Hashiguchi and Ueno, 1977; Hashiguchi, 1978; Topolnicki, 1990; Kohgo et al., 1993; Asaoka et al., 1997; Hashiguchi and Chen, 1998; Hashiguchi et al. 2002; Khojastehpor and Hashiguchi, 2004a,b; Hashiguchi and Tsutsumi, 2006; Hashiguchi and Mase, 2007; Wongsaroj et al., 2007). Consequently, its capability has been verified widely.

nˆ '

ıD 'n ı

0

ıD ' ıD 't

Subloading surface

σ ij'

Normal- yield surface Fig. 7.8 Normal and tangential stress rates for subloading surface model in deviatoric stress plane

7.6 Incorporation of Tangential Inelastic Strain Rate σ1

189

Yield surface

ıD

0

σ3

σ2

Input : ıD σ1

σ1

Normal- yield surface

Dt

Yield surface

Dt

0

0

σ3

σ2

σ2

Subloading surface model

Fig. 7.9 Incorporation of tangential inelastic strain rate

σ3 Conventional plasticity model: Rudnicki and Rice model

Chapter 8

Cyclic Plasticity Model: Extended Subloading Surface Model 8 Cyclic Pla sticity Model: Extended Sublo ading Surface Model

The subloading surface model described in Chapter 7 would be the only pertinent unconventional model for the description of monotonic loading behavior but it cannot describe the cyclic loading behavior pertinently by the formulation itself shown therein, predicting the unrealistically large plastic strain accumulation because only elastic deformation is predicted in the unloading process. Various unconventional plasticity models aimed at describing the cyclic loading behavior, i.e. the cyclic plasticity models have been proposed, while models other than the extension of the subloading surface model involve serious defects individually. The basic features of these models will be reviewed first and thereafter the formulation of the extended subloading surface model is described in detail in this chapter.

8.1 Classification of Cyclic Plasticity Models Various cyclic plasticity models have been proposed to date. They are classifiable into the two types. One type is based on the idea of kinematic hardening, i.e., the translation of subyield surface assumed inside the yield surface: the size retains a constant ratio to the yield surface. The other type is based on the idea of a subloading surface which expands/contracts keeping similarity to the yield surface as the stress approaches/leaves the yield surface described in the last section, in which a single independent surface is assumed. The classification is shown schematically in Fig. 8.1.

8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening The description of plastic deformation induced in the subyield state has been initiated through the extension of kinematic hardening concept as described below. 8.2.1 Multi-Surface Model Mroz (1967) and Iwan (1967) proposed the multi surface model which assumes the multiple encircled subyield surfaces which are pushed out by the current stress, K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 191–209. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

192

8 Cyclic Plasticity Model: Extended Subloading Surface Model

Cyclic plasticity model

Translation

Expansion/contraction

of subyield surface

of loading surface

Linear kinematic - hardening rule

Multi surface model Mroz (1966, 1967) Iwan (1967) Mathematical refinement

Infinite surface model Mroz et al. (1981)

Nonlinear kinematic - hardening rule

Nonlinear expansion rule

Two surface model

Subloading surface model

Dafalias & Popov (1975) Krieg (1975)

Hashiguchi-Ueno (1977) Hashiguchi (1980)

Single small yield surface Combination of several kinematic hardening rules

Movement of Similarity-center

Single surface model: Nonlinear kinematic hardening model

Extended subloading surface model

Armstrong-Fredericson (1966) Chaboche et al. (1979)

Hashiguchi (1986, 1989)

Fig. 8.1 Classification of cyclic plasticity models

retaining the ratios of their sizes to the size of the yield surface as constant. The plastic modulus depends on the ratio for the subyield surface on which the current stress lies. In this model, the stress translates by the difference of the half sizes of subyield surfaces and then contacts with a larger subyield surface in order in the initial loading process from the initial isotropic state where the subyield surfaces are concentric with respect to the origin of stress space. On the other hand, it translates by the difference of the sizes of subyield surfaces, obeying the linear kinematic hardening rule, and then contacts with a larger subyield surface in turn in the unloading-reverse loading process. Therefore, the Masing rule (Masing, 1926) meaning that the curvature of stress-strain curve in the unloading-reverse loading decreases compared to the curvature of initial loading curve is described exactly and simply. On account of this mechanical feature, this model has been used widely. However, it includes the following serious defects. 1) Variations in curvatures of stress-strain curves in test data are not as great as those described by this model. 2) Subyield surfaces with different sizes having different plastic moduli contact mutually at one point. Then, the singular point of plastic modulus is induced at the contact point. Numerical calculation of cyclic loading behavior in the vicinity of contact point becomes unstable.

8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening

σ

193

Normal-yield state Constant stress amplitude No strain accumulation (Trace of fixed loop) Elastic state

εp

0 Nonh-hardening state

σ

Normal-yield state

Constant stress amplitude No strain accumulation (Gradual contraction of loop) Elastic state

0 Hardening state

εp

Fig. 8.2 Prediction of cyclic loading behavior by the multi surface model under constant one-side stress amplitude

3) Plastic modulus decreases suddenly at the moment when the stress reaches a larger subyield surface. Therefore, the smoothness condition (Hashiguchi, 1993a, b, 1997, 2000) is violated at that moment. Needless to say, the smooth stress-strain curve cannot be described. 4) Plastic strain accumulation in cyclic loading cannot be described for the stress amplitude within the smallest subyield surface enclosing a purely elastic domain. 5) In the cyclic loading process with the constant amplitude of the positive or negative one side stress, the plastic shakedown is induced immediately for non-hardening (sub)yield surfaces and after several cycles for hardening (sub)yield surfaces, tracing the fixed loop cyclically (Fig. 7.2). In other words, the accumulation of plastic strain for a cyclic loading under positive or negative one side stress amplitude, called the mechanical ratcheting effect, cannot be described at all by this model because of the linear kinematic hardening rule. Therefore, the deformations of machinery and structures subjected to cyclic loading are predicted to be unrealistically small.

194

8 Cyclic Plasticity Model: Extended Subloading Surface Model

6) The continuity condition (Hashiguchi, 1993a, b, 1997, 2000) is also violated at the moment when the stress transfers to a larger subyield surface if the tangential inelastic strain rate described in 6.6 is incorporated. 7) Judgment on which subyield surface among multiple subyield surfaces the current stress lies is required in the loading criterion. In addition, deformation analysis by this model is complicated because it is necessary to calculate all movements of multi-subyield surfaces. It increases memory usage and calculations necessary for numerical analysis. It becomes more serious in the analysis of cyclic loading behavior. 8.2.2 Two-Surface Model Dafalias and Popov (1975) and Krieg (1975) proposed the simplification of the multi surface model by assuming only one subyield surface enclosing a purely elastic domain. Then, it assumes two surfaces, i.e. the yield and the subyield

σ No strain accumulation (Elastic deformation)

Elastic state

εp

0

Cyclic loading inside subyield surface.

N orm al-y ield sta te

σ

Constant stress amplitude

Elastic state

εp

0 Constant one-side stress amplitude

Fig. 8.3 Prediction of cyclic loading behavior by the two surface model under constant one-side stress amplitude

8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening

195

surfaces and thus it is called the two surface model, while the yield surface is called the bounding surface by Dafalias and the limit surface by Krieg as described in 7.4. The plastic modulus is assumed to depend on the distance from the current stress on the subyield surface to the conjugate stress on the yield surface, resulting in the nonlinear kinematic hardening rule, whereas the outward normal of these surfaces have the same direction. Here, it is required that the subyield surface must translate so as not to intersect with the yield surface. The pertinent translation rules have been given by Hashiguchi (1981, 1988). This model have been adopted widely for the prediction of deformation behavior of metals (cf. e.g. Dafalias and Popov, 1976; McDowell, 1985, 1989; Ohno, 1982; Ohno and Kachi, 1986; Ellyin, 1989; Hassan and Kyriakides, 1992; Yoshida and Uemori, 2002, 2003). This model is accompanied with the defects 2), 3), 4), 6) and a part of 7 ) described for the multi surfaced model. Further, the plastic modulus is determined only by the distance from the current stress to the yield (bounding) surface in this model. Then, the stress-strain curve with a unique curvature is predicted independent of the loading processes in this model in direct opposition to the multi surface model. In other words, the curvatures of the initial, the reloading and the unloading-inverse loading curves cannot be distinguished. Therefore, the open hysteresis loop is predicted in the unloading-reloading process. On account of this feature, this model cannot predict the cyclic loading behavior pertinently, describing excessively large accumulation of plastic strain as shown in Fig. 8.3. The single surface model is proposed by Dafalias and Popov (1977), in which the subyield surface shrinks to a point in the two surface model. It would describe a smooth response and the dependence of the direction of plastic strain rate on the direction of stress rate but it would not be applicable to the cyclic loading behavior, predicting the open hysteresis loop. 8.2.3 Infinite-Surface Model Modification of the multi surface model was proposed by Mroz et al. (1981), in which infinite number of subyield surfaces are incorporated inside the yield surface in contrast to the two surface model. It is called the infinite surface model. The smoothness condition is fulfilled in the monotonic loading process but it is violated at the moment when the stress passes through the starting point of unloading in the reloading process after the partial unloading, whereas the plastic modulus is singular at that point, since subyield surfaces with different sizes contact mutually. All the defects 1)-7) described in the multi surface model except for the fulfillment of smoothness condition in the monotonic loading process are retained in the infinite surface model. 8.2.4 Nonlinear Kinematic Hardening Model The nonlinear kinematic hardening model proposed by Armstrong and Fredericson (1966) quite overestimates the mechanical ratcheting effect as the two surface model does. In order to improve this defect, the superposition of several nonlinear

196

8 Cyclic Plasticity Model: Extended Subloading Surface Model

σ

Experiment

½ ¾ Prediction: ¿ (no strain accumulation)

Elastic state Elastoplastic state

0

ε

Fig. 8.4 Prediction of cyclic loading behavior after slight unloading by the multi, the two and the nonlinear kinematic hardening models having a purely elastic domain

kinematic hardening rules was proposed by Chaboche et al. (1979). The mathematical structure seems quite simple at first view, while it is one of polynomial approximation so that a fitting to test data might be easy. On the other hand, it needs to use many material constants without clear mechanical meaning. The mechanical behavior of this model is identical to that of the two-surface model as examined by Chaboche (1989). A singularity of the plastic modulus is not induced because only one yield surface is assumed; however, the smoothness condition is violated when the stress reaches the yield surface. Therefore, similar defects as those in the two-surface model are involved in this model. Needless to say, the continuity condition is also violated if the tangential inelastic strain rate is incorporated. Eventually, this model is physically and mathematically quite irrational contrary to the repeated long reviews by Chaboche (1989, 2008) and Lamaitre and Chaboche (1990). A small (sub)yield surface enclosing the elastic domain is assumed in all the unconventional plasticity models described above. Then, the accumulation of plastic strain observed in experiments shown in Fig. 8.4 cannot be described for cyclic loading of partial unloading-reloading even immediately after the saturation of hardening. Therefore, it results in risky mechanical designs.

8.3 Extended Subloading Surface Model As described in the beginning, the subloading surface model formulated in Chapter 7, called the initial subloading surface model hereinafter, is incapable of describing cyclic loading behavior appropriately, predicting an open hysteresis loop in an unloading-reloading process and thus overestimating a mechanical ratcheting effect.

8.3 Extended Subloading Surface Model

197

This defect has been remedied by making the similarity-center of the normal-yield and the subloading surfaces move with the plastic strain rate after Hashiguchi (1986, 1989) as will be explained below. Now, adopt the normal-yield surface with the anisotropy in Eq. (6.84), i.e. f (σˆ ,β ) = F ( H )

(8.1)(re-record)

for the formulation of the extended subloading surface model. Denoting the similarity-center by the symbol s of the normal-yield and the subloading surfaces, the conjugate point in the subloading surface for the back stress α in the normal-yield surface by α , the conjugate point on the normal-yield surface for the current stress σ in the normal-yield surface by σ y and the ratio of the size of extended subloading surface to that of the normal-yield surface by following relations hold (see Fig. 8.5).

σ = R σˆ y

(8.2)

s = Rsˆ, α = s − Rsˆ where

σˆ y ≡ σ y − α

sˆ ≡ s − α

R , the

，σ ≡ σ −α

，s ≡ s−α

(8.3) (8.4) (8.5)

It holds from Eqs. (8.3) and (8.5)2 that

σ ≡ σ + Rsˆ

(8.6)

where

σ ≡ σ −s

(8.7)

Regarding σ as σ y and substituting Eq. (8.2) into Eq. (8.1) of the normal-yield surface, the expression of the subloading surface is given as f (σ, β) = RF ( H )

(8.8)

By substituting Eq. (8.6) into Eq. (8.8), the subloading surface is rewritten as follows: f (σ + Rsˆ, β) = RF ( H )

(8.9)

The normal-yield ratio R can be calculated from Eq. (8.9), while there does not exist an analytical solution in general. The explicit equations and calculation methods for metals and soils will be shown in Chapters 10 and 11, respectively.

198

8 Cyclic Plasticity Model: Extended Subloading Surface Model

−

N

− N

ıy

ıˆ y (=

− ı−/R ) Normal-yield surface

ı

− ı

Subloading surface

− Į

Į

sˆ s

σ ij

0

Fig. 8.5 Normal-yield and subloading surfaces

The material-time derivative of Eq. (8.8) leads to

t r(

• ∂f (σ , β ) ∂f ( , β ) ∂f ( , β ) • σ) − t r( σ α) + t r ( σβ β) = R F + RF ′ H ∂σ ∂σ ∂

(8.10) where it holds from Eq. (8.3) that •

α = R α − R sˆ + (1 − R ) s

(8.11)

If the similarity-center s is located beyond the normal-yield surface, the subloading surface intersects with the normal-yield surface, resulting in the impertinence that the outward-normal of the subloading surface differs from that of the normal-yield surface in the intersecting point. Therefore, the similarity-center must lie inside the normal-yield surface. Consequently, the following inequality must hold.

f (sˆ, β) ≤ F ( H ) The time-differentiation of Eq. (8.12) at the limit state that normal-yield surface (see Fig. 8.6) leads to

(8.12)

s

lies on

∂f (sˆ, β) ∂f (sˆ, β) − F• ≤ 0 for f (sˆ, β) = F ( H ) t r{ ( s − α )} + t r ( β) ∂β ∂ sˆ

8.3 Extended Subloading Surface Model

199

which is rewritten as ∂f (sˆ, β) ∂f (sˆ, β) ∂f (sˆ, β) sˆ}t r ( β β) t r{ ( s− α)} + 1 t r { F ˆ ˆ ∂ s s ∂ ∂ −

• ∂f (sˆ , β ) 1 t r{ sˆ}F ≤ 0 for f (sˆ, β) = F ( H ) ˆ F ∂s

(8.13)

noting the relation t r{(∂f (sˆ, β) / ∂ sˆ) sˆ} = F due to the Euler’s homogeneous function of degree-one. Further, Eq. (8.13) is rewritten as tr

[ ∂f (∂sˆs,ˆ β) ( s− α + F1 {t r ( ∂f (∂sˆβ, β) β) − F}sˆ ) ] ≤ 0 •

for f (sˆ , β ) = F ( H )

(8.14)

Equations (8.12) and (8.14) are called the enclosing condition of similarity-center. Let the following equation be assumed, which fulfills Eq. (8.14) since ∂f (sˆ) / ∂ sˆ and σ y − s produces the obtuse angle as shown in Fig. 8.6.

− N

− N

•

ıy

− ı /R

ıˆ y

•

ı − ı

∂ f ( sˆ, ȕ) ∂ sˆ

Ds ı

s

•Į−

sˆ

•

Į

Subloading surface

σ ij

0

Normal- yield surface Fig. 8.6 Limit state that similarity-center lies on normal-yield surface

200

8 Cyclic Plasticity Model: Extended Subloading Surface Model

s− α +

1 ∂f (sˆ, β) β { ) − F• }sˆ = c || D p||σ y tr ( F ∂β

(8.15)

where

σy ≡ σy − s

(8.16)

and c is the material constant. The evolution rule of the similarity center is given from Eq. (8.15) as follows: • s = c|| D p|| σ + α + { F ' H − 1 tr ( ∂f (sˆ, β) β)}sˆ F F β

(8.17)

∂

R

Although Eq. (8.17) is formulated so as to avoid the protrusion of stress from the normal-yield surface, it is conceivable that s approaches the current stress in general • as shown for the non-hardening case, i.e. α = β = 0 and H = 0 in Fig. 8.7. The close approach of s to the normal-yield surface can be avoided by replacement of σ / R to σ / R − sˆ / ℜ M ( ℜ M (< 1) : material constant prescribing the maximum value of ℜ s ≡ f (sˆ, β) / F ) in Eq. (8.17) when it is required (cf. Hashiguchi, 1986, 1989).

ıy

DD

ss

ı

•

•

s − Į •

Į • Subloading surface

0

Normal-yield surface

σ ij

Fig. 8.7 Direction of translation of the similarity- center in the general state (illustration for • non-hardening state: F = 0, α = β = 0 )

8.3 Extended Subloading Surface Model

201

sa F

sa ( F − sa 0 ) 1 − exp{∓ c 3 (ε ap − ε ap0 )} 2

[

c 3 ( F − sa )

] sa 0

0

1 ε ap0

−c

2

3 ( F + sa ) 2

−1

ε ap

ε ap

F

Fig. 8.8 Similarity-center vs. plastic strain curve in uniaxial loading process for non-hardening state

For the non-hardening case the relation of the axial components sa and Dap of

s and

D p is given from Eq. (8.17) as follows:

sa = c 3 (± Dap )(± F − sa ) (upper: Dap > 0, lower: Dap < 0) 2

(8.18)

in the uniaxial loading process under the plastically-incompressible state. The time-integration of Eq. (8.18) is given as

sa = (± F − sa 0 )[1 − exp{∓ c 3 (ε ap − ε ap0 )}] + sa 0

(8.19)

2

which is shown in Fig. 8.8. Substituting Eq. (8.17) into Eqs. (8.11), one has •

•

α = R α + (1 − R )[c|| D p || σ + α + { FF − 1 t r ( ∂f (sˆ, β ) β)}sˆ ] − R sˆ F R ∂β (8.20)

202

8 Cyclic Plasticity Model: Extended Subloading Surface Model

and further substituting Eq. (8.20) into Eq. (8.10), it follows that tr (

∂f (σ, β) σ) ∂σ

[∂f (∂σσ, β){R α + (1 − R )[c|| D || Rσ + α ∂f (σ, β ) β) = R F + R F + { F − 1 tr ( ∂f (sˆ, β) β)}sˆ ] − R sˆ}] + t r ( ∂β F F β ∂ − tr

p

•

　

•

•

•

which, using the relation N≡

∂ f (σ , β ) ∂σ

∂f (σ, β ) = ∂σ

tr

∂ f (σ , β ) ∂σ

( ∂f (∂σσ, β ) σ) tr(Nσ)

N=

(8.21)

( N = 1)

f (σ, β ) RF N= N tr(Nσ) tr(Nσ)

(8.22)

based on the Euler’s theorem for the homogeneous function in degree-one, is rewritten as follows: RF tr( N σ ) tr( Nσ ) − RF tr N α + (1 − R ) c|| D p || σ R tr( Nσ )

[{

[

•

• ˆ + {F − 1 tr ( ∂f (s, β) β)}sˆ − R sˆ F F ∂β

•

}] + tr( ∂f (∂σβ, β) β) = R F + R F

]

•

i.e. •

∂f (sˆ, β) F 1 β)}sˆ ] tr(N σ ) − tr N α + (1 − R )[c|| D || σ +{ − tr ( F ∂β F R

[{ •

p

+ tr N

•

•

1 tr( ∂f (σ, β ) β) − R − F R F ∂β RF

}] [ {

− R sˆ

}σ] = 0

i.e. •

F {σ + (1 − R ) sˆ} + α + • ( 1 σ − sˆ ) + c( 1 − 1)|| D p ||σ R tr( N σ ) − tr N F R R

[{

∂f (σ, β ) ∂f (sˆ, β) − − 1 tr ( β)sˆ β )σ − 1 R tr ( RF ∂β ∂β F

}] = 0

8.3 Extended Subloading Surface Model

203

This equation, with the relations

σ + (1 − R ) sˆ = σ − α + s − α − (s − α) = σˆ ⎫ ⎪ ⎬ ⎪ ⎭

1 σ − sˆ = 1 ( + ˆ ) − sˆ = 1 σ σ Rs R R R on account of Eqs. (8.4) and (8.6), is transformed as

[{

•

•

R 1 tr(N σ) − tr N F σˆ + α + σ + c( − 1)||D p ||σ F R R

∂f (σ , β ) ∂f (sˆ , β) − − 1 tr ( β)sˆ β )σ − 1 R tr ( ∂β RF ∂β F

，

}] = 0

Substituting Eqs. (6.37) (6.86) and (7.13), regarding R as equation, the following consistency condition is obtained.

(8.23)

R , into this

[ { FF ′h(σ, H ; D )σˆ + a (σ, H )||D ||

tr(N σ) − tr N

i

p

i

p

( ) 1 +{ U R + c( − 1)}|| D p||σ − 1 tr( ∂f (σ, β ) b (σ, H i )) || D p' ||)σ R R RF ∂β ∂f (sˆ, β) − b (σ, H i )) || D p' ||)sˆ − 1 R tr ( ∂β F

}] = 0

(8.24)

Substituting the associated flow rule D p = λ N (λ > 0)

(8.25)

into Eq. (8.24), it holds that

[ { FF ′h( σ, H ; N) σˆ + a(σ, H )

tr(N σ) − λ tr N

i

i

U ( R ) + c 1 − 1) σ − 1 tr( ∂f (σ, β ) b (σ, H ) || ||)σ +{ ( R } RF i ) N' ∂β R

−

∂f (sˆ, β) 1− R b (σ, H i )) || N' ||)sˆ tr ( ∂β F

}] = 0

from which one has

λ=

tr(N σ) , Mp

Dp =

tr( N σ ) N Mp

(8.26)

204

8 Cyclic Plasticity Model: Extended Subloading Surface Model σ

σ

Normal- yield surface

σ

σ

s α σ

σ s

σ s εp

0

0

εp

0

σ

Subloading surface: σ expands

σ

σ s

σ s

α

(d)

(a) σ

s

Expands σ 0

α εp

0

s α

α σ

0 0

(s =0) σ

εp

Contracts ( D p = 0)

0

(e) (b) σ

s

(s = 0)

σ σ

s

α

σ s

α

εp

0

σ

σ

σ

σ s

α Contracts (D p = 0)

0

α ε

0

Expands

p

0

(f)

(c)

Fig. 8.9 Prediction of uniaxial loading behavior by extended subloading surface model: (a) initial state, (b) initial loading process, (c) unloading process until similarity-center, (d) unloading-inverse loading process after passing similarity-center, (e) reloading process until similarity-center and (f) reloading process after passing 䇭䇭 Similarity-center, similarity-center. (

䇭䇭䇭䇭 Center of 䇭䇭䇭 Stress, subloading surface)

where

[ {FF ′h( ı, H ; N) ıˆ + a(ı, H ) +{U R(R) + c( R1 − 1) }ı

M p ≡ tr N

−

|| N' || RF

i

i

( t r( ∂f (∂ıȕ, ȕ) b (ı, H ))ı + R(1 − R)tr( i

∂f (sˆ, ȕ ) b (ı, H i ))sˆ ∂ȕ

)}]

(8.27) The strain rate is given from Eqs. (6.28), (6.29) and (8.26) as D = E −1 σ + t r( Npσ ) N M

(8.28)

from which the proportionality factor described in terms of the strain rate, denoted by Λ instead of λ , in the flow rule (8.25) is given as follows: Λ=

t r (NED) M p + t r (NEN)

(8.29)

8.4 Modification of Reloading Curve

205

Using Eq. (8.29), the stress rate is given from Eq. (8.28) as follows:

σ = ED −

t r (NED) EN M p + t r (NEN)

(8.30)

The loading criterion is given by

⎫ ⎪ ⎬ p D = 0 : otherwise ⎪ ⎭ Dp ≠ 0 : Λ> 0,

(8.31)

The relation of axial stress vs. plastic strain is illustrated for the uniaxial loading process of non-hardening material in Fig. 8.9. The closed hysteresis loop is depicted in this figure.

8.4 Modification of Reloading Curve The relation R − R0 = f (ε p − ε 0p ) holds in the loading process from Eq. (7.13) with the function U of only normal-yield ratio R , where it is set that

σ

n ic M o n oto g in d a lo

in Re lo a d

g

−

−

R =1

Rb − Ra

s

(similarity - center)

p ε a~b

p ε a~b

εp

Same Fig. 8.10 The defect of past subloading surface model: Unrealistically gentle returning to monotonic loading curve

206

8 Cyclic Plasticity Model: Extended Subloading Surface Model

−

σ

R =1

s (similarity-center)

εp

Modification

−

σ

R =1

s (similarity-center)

εp

Fig. 8.11 Description of cyclic loading behavior in the neighborhood of yield surface

and R 0 and ε 0p are the initial values of R and ε p, and thus the inverse relation is given by ε p − ε 0p = f −1 ( R − R0 ) . Therefore, the accumulation of the p ε p ≡ ∫ || D || dt ,

magnitude of plastic strain rate for a certain change of R is identical independent of loading processes, i.e. the initial loading process, the reloading process after a complete or partial unloading and the inverse loading process. This results in the prediction that the returning of the reloading stress-strain curve to the monotonic loading curve is unrealistically too gentle as shown in Fig. 8.10. In particular, the excessively large plastic strain accumulation is predicted for the cyclic loading process in the neighborhood of yield surface as shown in Fig. 8.11. It can be stated that the curvature of unloading-inverse loading curve is smaller than that of the initial loading curve but the curvature of reloading curve is larger than it as has been described in the Masing rule (Masing, 1926). In order to describe this behavior, let the material constant u in Eq. (7.13) for the evolution rule of the normal-yield ratio be extended as follows:

8.4 Modification of Reloading Curve

207

1 ) u is more different from the value in the initial isotropic state as the similarity-center is nearer to the normal-yield surface. In order to describe the approaching degree to the normal-yield surface, supposing the similarity-center surface which passes through the current similarity-center and is similar to the normal-yield surface, the similarity-center yield ratio ℜ s describing the ratio of the size of the similarity-center surface to that of the normal-yield surface is introduced. The similarity-center surface is described by the equation f (sˆ, β) = ℜ s F ( H ) by the replacement σ → s, α → α, R → ℜ s in the yield surface (6.84). Then, the similarity-center yield ratio is described by ℜ s = f (sˆ, β) / F ( H ) (0 ≤ ℜ s ≤ 1)

(8.32)

in terms of the known variables s, α, β, F . 2 ) The similarity-center yield ratio is large and the deviatoric stress lies outside of the similarity-center surface in the reloading process. Conversely, the similarity-center yield ratio is also large but the deviatoric stress lies inside of the similarity-center surface in the inverse loading process. Which the deviatoric stress lies outside or inside of the similarity-center surface can be judged by the sign of the following variable Sσ .

(

Sσ ≡ tr nˆ s

σr

|| σ r ||

) (− 1 ≤ Sσ ≤ 1)

(8.33)

⎛ ⎧ ⎞ ⎜ ⎪ ± 3 σ m (= ± σ m for σ' = 0) for nˆ s = ± 1 I ⎟ |σm | ⎜ ⎪ || σ || 3 ⎟ ⎜= ⎨ ⎟ ⎜ ⎪ ⎟ σ ' ˆ n ⎜⎜ ⎪ tr ( s || || ) for tr nˆ s = 0 ⎟⎟ σ ' ⎝ ⎩ ⎠

where ⎛ ⎧ 1 ⎞ I⎟ ⎜ ⎪σ for nˆ s = ± 1 3 ⎟ ˆ n + tr σ r ≡ σ' s σ mI ⎜ = ⎨ 3 ⎜ ⎪σ for tr nˆ s = 0 ⎟ ⎝ ⎩ ' ⎠ f (sˆ, β ) nˆ s ≡ ∂ ∂s

/ || ∂f (∂ss, β)|| ˆ

(8.34)

(8.35)

designates the normalized outward-normal of the similarity-center surface at the current similarity-center. The variable Sσ would be applicable to the general deformation behavior extending from the pressure-independent metals ( nˆ s = 0) to the isotropic consolidation ( nˆ s = ± I 3 ) in soils. nˆ s

Introducing these variables, let the material parameter modified as follows:

u in Eq. (7.13) be

⎛ ⎧u 0 exp(us ) for ℜ s = 1 and Sσ = 1 ⎞ ⎜ ⎪ ⎟ u = u 0 exp(usℜ s Sσ ) ⎜ = ⎨ u0 for ℜ s = 0 or Sσ = 0 ⎟ ⎜ ⎪u 0 exp( −us ) for ℜ s = 1 and Sσ = −1⎟ ⎝ ⎩ ⎠

(8.36)

208

8 Cyclic Plasticity Model: Extended Subloading Surface Model

u >u 0 (improved)

− R =1

−

σ

u =u 0

s (similarity - center)

εp

s u < u0 (improved) u =u 0 Fig. 8.12 Stress-plastic strain curve predicted by the modified subloading surface model: Rapid recovery to preceding monotonic loading curve

where u 0 and us are the material constants, while the former denotes the mean value of u . Then, u increases in the loading direction but inversely it decreases in the opposite direction. By this modification, the phenomenon that in the reloading curve after a partial unloading the stress returns rapidly to the monotonic loading curve can be described realistically as shown in Fig. 8.12. Then, the plastic strain accumulation for the cyclic loading process in the neighborhood of yield surface is suppressed as shown in Fig. 8.11.

8.5 Incorporation of Tangential-Inelastic Strain Rate Incorporating the tangential inelastic strain rate into Eqs. (8.28) and (8.30), the strain rate and the stress rate are given as follows:

ξ τ D = E−1 σ + tr(Npσ ) N + R σ't 2G M

σ = ED −

2G ξ R τ t r ( NED) EN − D't M + t r ( NEN) 1+ ξ Rτ p

(8.37) (8.38)

8.5 Incorporation of Tangential-Inelastic Strain Rate

209

where ⎫ ⎪⎪ ⎬ σ't ≡ I' σ = σ' − σ'n (tr(N σ't ) = 0) ⎪⎪ ⎭

σ'n ≡ n' ⊗ n' σ = tr(n' σ ) n'

(8.39)

D't ≡ I' D = D' − tr(n' D' ) n'

(8.40)

n' ≡ N' (|| n'|| =1) ||N' || I' ≡ I ' − n' ⊗ n'

(8.41) (8.42)

Chapter 9

Viscoplastic Constitutive Equations 9 Viscoplastic Co nstit utive Equatio ns

Deformation of solids exhibits the dependence on the rate of loading or deformation, i.e. the time- or rate-dependence. However, that dependence was deliberately ignored in preceding chapters. The history of development of the viscoplastic constitutive equation describing the rate-dependence of plastic deformation induced in the state of stress over the yield surface is first reviewed briefly. Then, the viscoplastic constitutive equations falling within the framework of the overstress model are described exhaustively, which would be applicable to the description of deformation for the wide range of strain rate from the quasi-static to the impact loads.

9.1 History of Viscoplastic Constitutive Equations Before pertinent formulation of the overstress model, the development of the viscoplastic constitutive equations is reviewed below. An overview of the history of the development of viscoplastic constitutive equations is portrayed in Fig. 9.1. The elastic constitutive equation extended so as to describe the rate-dependence is called the viscoelastic constitutive equation and one of the typical models is the Maxwell model, in which the spring and the dashpot are connected in series. Therefore, the strain rate ε• is additively decomposed into the elastic strain rate ε• e = E −1 σ• and the viscous strain rate ε• v = μ −1σ , where σ designates the stress, E is the elastic modulus and μ is the viscous coefficient. This model is concerned with the rate-dependent deformation at the low stress level below the yield stress. On the other hand, the elastoplastic constitutive equation can be schematically expressed by the Prandtl model in which the dashpot is replaced with the slider in the Maxwell model, whereas the slider begins to move in the state that the stress σ goes over the yield stress σ y , by which the plastic strain rate is induced. Furthermore, the model which describes the rate-dependent plastic strain rate ε• vp induced for the state of stress over the yield stress, called the viscoplastic strain rate, was introduced by Bingham (1922), assembling the above-mentioned Maxwell model and Prandtl model so as to connect the dashpot and the slider in parallel as shown in Fig. 9.1, where μ is the viscoplastic coefficient and n is the material constant. The Bingham model is the origin of the overstress model based on the concept that the viscoplastic strain rate is induced for the stress over the yield stress. K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 211–220. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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9 Viscoplastic Constitutive Equations

Maxwell’s viscoelastic model

μ

E

σ,ε

σ, ε •

•e

•v

−1 •

ε = ε +ε = E σ+

μ −1

σ

+ Prandtl’s elastoplastic model E

σ,ε

σ, ε

σy

⎧

•e −1 • ⎪ε = E σ for σ < σ y ε• = ⎨ ⎪ε• e + ε• p = E −1 σ• + M p −1σ• for σ = σ y

⎩

Bingham’s viscoplastic model (Bingham, 1922): Overstress model for one-dimension − μ

E

σ,ε

σ, ε

σf

ε• = ε• e + ε• v p = E −1 σ• + μ− −1< σ − σ f > n

Prager’s overstress model (Prager, 1961) 1961):: Overstress model for Mises metals

− −1 D = De + Dvp = E−1 σD + μ 〈

σ e − 1 〉n σ'

F (ε p e )

||σ' ||

General overstress model (Perzyna,1963): Overstress model for general materials

− −1〈 f (σ ) − 1 n N D = De + Dvp = E −1 σD + μ F (H ) 〉 Fig. 9.1 History of viscoplastic model

The above-mentioned Bingham model for the one-dimensional deformation was extended by Prager (Hohenemser and Prager, 1932; Prager, 1961a) to describe the three-dimensional deformation of metals, adopting the Mises yield condition as

9.1 History of Viscoplastic Constitutive Equations

213

shown in Fig. 9.1. In this model, the strain rate D is additively decomposed into the elastic strain rate De and the viscoplastic strain rate D vp, i.e. D = D e + D vp

(9.1)

with the viscoplastic strain rate D vp D vp = μ − 1 〈

σ e − 1 〉n σ'

F (ε v p )

|| σ' ||

(9.2)

where ε vp ≡ 2 / 3 ∫||Dvp' ||dt is the equivalent viscoplastic strain given by replacing the plastic strain rate D p to the viscoplastic strain rate D vp in the plastic equivalent strain ε pe ≡ 2 / 3 ∫|| D p' || dt . Furthermore, the viscoplastic strain rate in the Prager’s overstress model was extended by Perzyna (1963, 1966) for materials having the general yield condition unlimited to the Mises yield condition as Dvp = μ −1〈

f (σ ) n −1 N F (H ) 〉

(9.3)

Then, substituting Eq. (6.29) and (9.3) into Eq. (9.1), we have f (σ ) n D = E −1 σD + μ −1〈 − 1〉 N F (H )

(9.4)

f (σ ) σD = ED − μ −1〈 ( H ) − 1 〉n EN F

(9.5)

and thus

where μ depends on stress, internal variables and temperature. The simplest yield condition f (σ ) = F ( H ) in Eq. (6.30) is used for sake of simplicity in explanation. Internal variables prescribing the variation of yield surface evolve with the inelastic strain rate. Then, the evolution rule of the isotropic hardening variable H is given as •

H = h (σ, Hi , Dvp )

(9.6)

by replacing the plastic strain rate Dp to the viscoplastic strain rate Dvp in the evolution rule of the isotropic hardening variable in Eq. (6.37) for the plastic constitutive equation. Here, the hardening function F ( H ) signifies the size of yield surface and f (σ ) does the size of the surface which passes through the current stress point and has similar shape and orientation to the yield surface, whereas the latter surface is called the dynamic-loading surface.

214

9 Viscoplastic Constitutive Equations

On the other hand, the creep model, which also aims at describing the inelastic deformation in the state of stress over the yield stress as well as the overstress model, has been studied (cf. Norton, 1929; Odqvist and Hult, 1962; Odqvist, 1966) by extending the yield condition such that the yield surface expands with the creep strain rate Dc as follows: || c|| f (σ ) = F ( H ) ( D c D0

) /m 1

(9.7)

where m ( 1) is the material constant and D0c is the reference creep strain rate. It is obtained from Eq. (9.7) that f (σ ) m (9.8) || Dc|| = D0c ( ) F (H )

In addition, adopting the associated flow rule, the creep strain rate is given by f (σ) m Dc = || Dc|| N = D0c ( F ( H ) ) N

(9.9)

and then the strain rate is given by f (σ) m D D = De + Dc = E−1 σ + D0c ( F ( H ) ) N

(9.10)

The creep model has different structures from the overstress model because it has no threshold value for the generation of the viscoplastic strain rate. Therefore, this model cannot describe appropriately the deformation behavior at a low rate since it does not reduce to the elastoplastic constitutive equation in a quasi-static deformation. In fact, it predicts the viscoplastic strain rate leading to the overshooting stress-strain curve at the moment of unloading in the quasi-static loading process. Consequently, this model is going to be disused and then it will be replaced over time by the overstress model. Furthermore, various constitutive models involving the time itself elapsed after a certain physical event begins/stops have been proposed to date. Here, note that the estimation of time when a physical event begins/stops depends on the subjectivity of observers, especially in a fluctuating rate of deformation. Therefore, these models are impertinent, lacking an objectivity.

9.2 Mechanical Response of Ordinary Overstress Model The development of rate-dependent elastoplastic constitutive equation is reviewed above and it is described that the overstress model would have a pertinent basic structure. Here, let the mechanical responses at the infinitesimal and the infinite rates of deformation be examined in order to clarify the basic property of this model.

9.3 Modification of Overstress Model: Extension to General Rate of Deformation

215

Equation (9.4) is rewritten using the notation dε ≡ Ddt as follows: dε = E−1dσ + μ −1 〈

f (σ ) n − 1〉 Ndt F (H )

(9.11)

In the infinitesimal rate of deformation, i.e. the quasi-static deformation state, Eq. (9.11) reduces to 0 ≅ 0 + μ −1 〈

f (σ ) n − 1〉 Ndt F (H )

(9.12)

Then, the stress varies fulfilling the yield condition f (σ ) = F ( H ) so that Eq. (9.12) approaches the response of the elastoplastic constitutive relation in the infinitesimal rate of deformation. On the other hand, in the infinite rate of deformation, Eq. (9.11) reduces to dε ≅ E −1dσ + 0

(9.13)

thereby approaching the elastic response. Therefore, it predicts the unrealistic response that the material can bear an infinite load. Eventually, the existing overstress model in Eq. (9.4) describes pertinently the deformation behavior in a low rate. However, it is inapplicable to the prediction of deformation at a high rate. The material constant n included as the power form in Eq. (9.11) is usually selected to be larger than five, but the fitting to the test data for impact load is impossible even if n is selected as one hundred which, needless to say, results in the inappropriate prediction of deformation in a slow loading process. In addition, the inclusion of a high power in the equation induces difficulty in numerical calculations.

9.3 Modification of Overstress Model: Extension to General Rate of Deformation The ordinary overstress model in Eq. (9.4) is inapplicable to the description of deformation at a high rate, as described above. We modify it to exclude this defect. Then, let the following equation be adopted instead of Eq. (9.3). D vp = C N

(9.14)

where C is the material function fulfilling the following conditions (see Fig. 9.2). ⎧= 0 for f (σ ) ≤ F ( H ) C⎨ ⎩→ ∞ for f (σ ) → R m F ( H )

(9.15)

R m is the material constant designating the maximum value of the ratio f (σ) / F ( H ) , whereas R m 1 would hold usually. The simplest equation fulfilling the condition (9.15) is given by

216

9 Viscoplastic Constitutive Equations

Modified overstress model ¢ f (ı ) − F ( H ) ² C = μ− −1 f m F ( H ) − f (ı )

C

Existing overstress model f (ı ) n −1 C = μ− −1¢ F (H ) ²

0

1

Rm F (H )

f (ı )

Fig. 9.2 Function C䇭 prescribing the magnitude of viscoplastic strain rate in the existing and the modified overstress models

C = μ −1

〈 f (σ ) − F ( H ) 〉 R m F ( H ) − f (σ )

(9.16)

Adopting Eq. (9.16) in Eq. (9.14), the viscoplastic strain rate is given as p DNv = μ −1

〈 f (σ ) − F ( H ) 〉 N R m F ( H ) − f (σ )

(9.17)

Then, it follows that 〈 f (σ) − F ( H ) 〉 −1 D = E σD + μ −1 R F ( H ) f (σ) N − m

(9.18)

〈 f (σ) − F ( H ) 〉 EN σD = ED − μ −1 R F ( H ) − f (σ) m

(9.19)

The stress-strain curves predicted by the existing overstress model (9.4) and the modified model (9.19) are depicted schematically in Fig. 9.3. As seen in this figure, the viscoplastic strain rate is induced infinitely as f (σ ) approaches R m F ( H ) in the modified model. Therefore, the defect in Eq. (9.4) that the stress increases elastically for the infinite rate of deformation is resolved in Eq. (9.19).

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model

217

Stress-strain curve for impact load ( || D || → ∞ ) predicted by the existing overstress model

σ f (ı )/ F ( H ) = R m

Impact load || D ||→ ∞

Stress-strain curve for impact load ( || D || → ∞ ) predicted by the modified overstress model

|| D|| increases

f (ı )/ F ( H ) = 1: yield state

Quasi-static load || D || ≅ 0

0

ε

Fig. 9.3 Stress-strain curve predicted by the existing and the modified overstress models

The necessity of incorporation of the plastic strain rate depending on the magnitude of strain rate in addition to the elastic and the viscoplastic strain rates has been inferred in order to describe the deformation at a high rate (cf. Lamaitre and Chaboche, 1990; Hashiguchi et al., 2005a, Chaboche, 2008). However, it induces the physical contradiction that the plastic strain rate is described twice in the two terms of the viscoplastic and the plastic strain rates in the quasi-static deformation process. In addition, inclusion of the magnitude of strain rate leads to the rate non-linearity of the constitutive equation, thereby causing difficulty in numerical calculations.

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model The overstress model formulated in 9.3 falls within the framework of conventional plasticity on the premise that the interior of yield surface is an elastic domain (Drucker, 1988). Consequently, it exhibits an abrupt transition from the elastic to the viscoplastic state, violating the smoothness condition. Moreover, it violates not only the smoothness but also the continuity conditions (Hashiguchi, 1993a, b, 1997, 2000) when the tangential inelastic strain rate is introduced. This defect in the

218

9 Viscoplastic Constitutive Equations

existing overstress model would be remedied by introducing the concept of the subloading surface (Hashiguchi, 1980, 1986, 1989). Explicitly, we would have only to modify such that the viscoplastic strain rate is induced by the expansion of the dynamic-loading surface from the subloading surface, whereas it is induced by that from the yield surface in the existing overstress model. Let the ratio of the size of dynamic-loading surface to that of normal-yield surface be called the dynamic-loading ratio. It is denoted by the same symbol R as the normal-yield ratio in the rate-independent subloading surface model, noting that the dynamic-loading surface passes through the current stress. On the other hand, let the ratio evolving by the following equation which is given by replacing the plastic strain rate to the viscoplastic strain rate in Eq. (7.13) for the rate-independent subloading surface model be called the subloading ratio, denoted by the symbol R s (0 ≤ R s ≤ 1) in stead of the normal-yield ratio R , and let the surface having this ratio to the normal-yield surface be again called the subloading surface. ⎧U ( R s )|| Dvp|| for Dvp ≠ 0 • ⎪ Rs = ⎨ • vp ⎪⎩ R (R s = R ) for D = 0

(9.20)

The stress-controlling ability attracting stress to normal-yield surface in the quasi-static deformation process is retained by incorporating Eq. (7.14) with the replacement of R to R s. The explicit equation for the function U ( R s) satisfying Eq. (7.14) can be given by the identical form as Eq. (7.15), i.e. U ( R s ) = u cot( π 2

〈 R s − Re 〉 ) 1 − Re

(9.21)

R s can be calculated analytically through integration of Eq. (9.20)1 with Eq. (9.21) in the viscoplastic deformation process Dvp ≠ 0 similarly to Eq. (7.16) as p p R s = π2 (1 − Re )cos −1{cos( π R0 − Re )exp( − π u ε − ε 0 )} + Re 2 1 − Re 2 1 − Re

(

9.22) under the initial condition ε p = ε 0p : R s = R 0. Now, on the formulation of the viscoplastic strain rate, let the following assumptions be incorporated. 1. 2. 3.

The viscoplastic strain rate is induced when the stress goes over the subloading surface. There exists the limit of the dynamic-loading ratio R and viscoplastic strain rate Dvp is induced infinitely as R reaches the limit value. The viscoplastic strain rate has the outward-normal direction of the dynamic loading surface.

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model

4.

219

Evolution rules of all the internal variables are given by the rules for the rate-independent subloading surface model with the replacement of the plastic strain D p into the viscoplastic strain rate Dvp .

Based on these postulates, let the viscoplastic strain rate be given by the following equation. Dv p = C ( R , R s) N

(9.23)

where C ( R, R s) is the monotonically-increasing function of R fulfilling the following conditions.

= 0 for R ≤ R s C ( R, R s) ® ¯→ + ∞ for R = R m

(9.24)

Rm signifies the limit value of the dynamic-loading ratio R . The function C ( R, R s) fulfilling Eq. (9.24) is schematically shown in Fig. 9.4. Here, the dynamic-loading ratio R is calculated simply by f (σ ) / F ( H ) for the initial

subloading surface model, but it must be calculated by Eq. (8.9) for the extended subloading surface model. The evolution rules of the internal variables H , α, β, s must be calculated considering the above-mentioned assumption 4. The simple equation fulfilling condition (9.24) is given as follows: 〈R − R 〉 C ( R , R s ) = μ −1 R − s m R

C ( R, R s)

(9.25)

Subloading-overstress model: ¢ R − Rs ² C ( R, R s) = μ− −1 Rm − R Existing overstress model: μ− −1 R − 1 n

¢

0

R e Rs 1

Rm

²

R

Fig. 9.4 Function C ( R, R s) prescribing the magnitude of viscoplastic strain rate in the subloading and the existing overstress models

220

9 Viscoplastic Constitutive Equations

Substituting Eq. (9.25) into Eq. (9.23), the viscoplastic strain rate is given as 〈R −R 〉 p Dv = μ −1 R − s N m R

(9.26)

Furthermore, substituting Eqs. (6.29) and (9.26) into Eq. (9.1), the strain rate is given as follows: 〈 R − Rs〉 (9.27) D = E−1 σD + μ −1 R m − R N from which the stress rate is described by 〉 〈 σD = ED − μ −1 RRm−−RRs E N

(9.28)

The stress-strain curve predicted by Eq.(9.28) is shown in Fig. 9.5. Incorporating the tangential inelastic strain rate formulated in 7.6 into Eq. (9.27), the strain rate and the stress rate are given as follows: D

D = E−1 σ + μ

−1 〈 R − R s 〉

+ ξ Rτ σD ' t Rm − R N

ıD = ED − μ −1 ¢RR −−RRs² E N − m

(9.29)

2G D't 1 1+ 2G ξ Rτ

whereas R must be replaced by R for the extended surface model.

: R= R m

σ

: maximum state

Impact load || D ||→ ∞

Rs =1

te : normal- yield sta

f (ı ) = Rm F ( H )

|| D|| increases

f (ı ) = F ( H )

Quasi-static loading || D|| ≅ 0 : f (ı ) = Rs F ( H )

0

ε

Fig. 9.5 Stress strain curve predicted by the subloading overstress model

(9.30)

Chapter 10

Constitutive Equations of Metals 10 Constitutive Equatio ns of Metals

The plasticity has highly developed through the prediction of deformation of metals up to date. The reason would be caused from the fact that, among various materials exhibiting plastic deformation, metals are used most widely as engineering materials and exhibit the simplest plastic deformation behavior without a pressure dependence, a plastic incompressibility, an independence of the third invariant of deviatoric stress and a softening. Thus, the elastoplastic constitutive equation of metals has developed to the highest level. Nevertheless, metals exhibit various particular aspects, e.g. the kinematic hardening and the stagnation of isotropic hardening in a cyclic loading. Explicit constitutive equations of metals will be delineated in this chapter, which have been formulated based on the general elastoplastic constitutive equations studied in the preceding chapters.

10.1 Isotropic and Kinematic Hardening The yield function for the Mises yield condition is extended to incorporate kinematic hardening by replacing σ' to σˆ ' in Eq. (6.54) as follows: f (σˆ ) = 3 || σˆ ' || 2

， Nˆ = Nˆ ' = nˆ' = || σσˆˆ '' ||

(10.1)

while the subloading function f (σ ) for Eq. (10.1) is given by f (σ ) =

3 || || 3 2 σ' (= 2 || σ' + R sˆ' || )

， N = N' = n' = || σσ'' ||

(10.2)

noting Eq. (8.6) or (8.9) with β = 0 . The explicit form of the hardening function (6.54) is given as follows

．

(Hashiguchi and Yoshimaru, 1995)

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 221–248. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

222

10 Constitutive Equations of Metals

F (ε p e) (1 + h1) F0

F' = F0 h1 h2 exp(−h2ε p e) 1

F0

0

ε pe

Fig. 10.1 Isotropic hardening function in the uniaxial loading process

F (ε p e ) = F0 [1 + h1{1 − exp(−h2 ε p e )}] F' = F0 h1 h2 exp(−h2 ε p e ) (10.3) where h1, h2 are the material constants. The hardening function F (ε p e ) in Eq. (10.3) increases from the initial value F0 with the plastic equivalent strain ε p e and saturates when it reaches the maximum value (1 + h1)F0 . The evolution rule of the back stress was given by Prager (1956) as follows:

α = ap Dp = ap || D p||Nˆ

(10.4)

where ap is the material constant having the dimension of stress. According to Eq. (10.4), the component of back stress is induced even in the direction of zero stress condition seen in the uniaxial loading process and in plane stress condition for instance. In order to avoid this inconvenience, Ziegler (1959) proposed the following equation

α = az ||Dp || σˆ

(10.5)

where a z is the dimensionless material constant. In Eq. (10.5) the mean component of α is induced and thus α does not become a deviatoric tensor. The translation directions of α in Eqs. (10.4) and (10.5) are same in the deviatoric stress plane for the Mises yield surface, but they generally differ from each other in the other yield surfaces as shown in Fig. 10.2. Which rule is more pertinent physically is not clarified yet.

10.1 Isotropic and Kinematic Hardening

223

ˆ N

ı ıˆ

D Prager : : Į Prager

D Ziegler : : Į Ziegler

Į σ ij

0

Yield surface

Fig. 10.2 Comparison of Prager’s (1955) and Ziegler’s (1959) kinematic hardening rules

According to Eqs. (10.4) and (10.5), the relation of back stress vs. plastic strain is linear when ap and a z are constant and thus they are called the linear-kinematic hardening rule. On the other hand, it is observed in test data that the rate of back stress decreases gradually and increases abruptly at the initiation of inverse loading. The nonlinear kinematic hardening rule taking account of these behavior was proposed by Armstrong and Fredericson (1966). The explicit equation is given as follows (Hashiguchi, 1989):

α = 32 aα ( rα F N − α)||D p || , a ≡ 32 aα ( rα F N − α)

(10.6)

where aα and rα (< 1) are material constants. As shown in Fig. 10.3, α translates to approach the conjugate point σ' = rα F N (= rα F σ ' / || σ ' ||) on the limit surface || σ ' || = rα F of kinematic hardening in the deviatoric stress plane, i.e. α → rα F σ ' / || σ ' || in Eq. (10.6). The relation of axial back stress α a vs. axial

logarithmic plastic strain ε ap is given from Eq. (10.6) as follows:

dα a = ± aα (± 2 / 3 rα F − α a ) d ε ap

(10.7)

where the upper and lower signs signify extension and compression, respectively. The integration of Eq. (10.7) under F = const. for sake of simplicity leads to the following equation.

224

10 Constitutive Equations of Metals

σ3

rα F ıˆ ' / || ıˆ ' ||

ı

ıˆ ĮD

Į

0

σ1

σ2 Kinematic hardeing limit surface || ı' || = rα F ( H )

Yield surface

ˆ ' || = F ( H ) 3/2 || ı

Fig. 10.3 Kinematic hardening rule (10.6) αa

2 / 3 rα F

2 / 3 rα F − α a

1

− ( 2 / 3 rα F + αa )

1

0

ε ap

− 2 / 3 rα F

Fig. 10.4 Kinematic hardening variable vs. plastic strain curve of (10.9) in uniaxial loading process

10.2 Cyclic Stagnation of Isotropic Hardening

225

r ε ap − ε ap0 = ∓ a1α ln ± 2 / 3rα F − α a ± 2/3 α F − αa0

(10.8)

where α a 0 and ε ap0 are the initial values of α a and ε ap , respectively. It is obtained from Eq. (10.8) that

α a = ± 2 / 3 rα F 1− (1 ∓

[

αa0 2 / 3 rα F

) exp{∓ k a (ε ap − ε ap )}] 0

(10.9)

which is depicted in Fig. 10.4.

10.2 Cyclic Stagnation of Isotropic Hardening The isotropic hardening of metals is induced by the equivalent plastic strain. It is observed in experiments that isotropic hardening stagnates despite of the development of the equivalent plastic strain for a certain range in the initial stage of re-yielding after reverse loading. This phenomenon remarkably influences the cyclic loading behavior in which the reverse loading is repeated. In particular, the isotropic hardening saturates finally in the cyclic loading under a constant strain amplitude. In order to describe this phenomenon the idea of the nonhardening region was proposed by Chaboche et al. (1979; see also Chaboche, 1989; Lemaitre and Chaboche, 1990) and modified by Ohno (1982) and Ohno and Kachi (1986): the isotropic hardening does not proceeds when the plastic strain exists in a certain region of the plastic strain space which expands with the cyclic loading. It is similar to the notion of the yield surface, assuming that the plastic strain is not induced when the stress lies inside it. Thereafter, the formulation that the isotropic hardening stagnates when the back stress lies in the certain region of stress space was proposed by Yoshida and Uemori (2002, 2003). However, it cannot describe the stagnation behavior of isotropic hardening of metals without the kinematic hardening. The following defects are imposed in these formulations. 1 ) The isotropic hardening is induced suddenly when the plastic strain or the back stress reaches the boundary of nonhardening region, violating the smoothness condition (Hashiguchi, 1993a, b, 1997, 2000). Thus, the smooth stress-strain curve cannot be described. 2 ) The judgment whether or not the plastic strain or the back stress reaches that boundary is required. 3 ) The numerical operation to pull back the plastic strain or back stress to that boundary so as not to go over the nonhardening region is required.

226

10 Constitutive Equations of Metals

In what follows, a pertinent formulation without these deficiencies is presented for the cyclic stagnation of isotropic hardening, based on the concept of subloading surface. Assuming that the isotropic hardening stagnates when the plastic strain ε p lies inside a certain region in the plastic strain space, let the following surface, called the normal-isotropic hardening surface be introduced.

f (εˆ p ) = K

(10.10)

where

εˆ p ≡ ε p − α

(10.11)

K and α designate the size and the center, respectively, of the normal-isotropic hardening surface, the evolution rules of which will be formulated later. The function f (εˆ p ) is explicitly given by

f (εˆ p ) =

2 || εˆ p || 3

(10.12)

Further, incorporate the surface, called the sub-isotropic hardening surface, which always passes through the current plastic strain point ε p and has similar shape and orientation to the normal-isotropic hardening surface (see Fig. 10.5). Then, it is mathematically expressed as

f (εˆ p ) = RK

(10.13)

where R (0 ≤ R ≤ 1) is the ratio of the size of sub-isotropic hardening surface to that of the normal-isotropic hardening surface and thus it plays the role as the measure to describe the approaching degree of plastic strain to the normal-isotropic hardening surface. Then, R is called the normal-isotropic hardening ratio. It is calculated from the equation R = f (εˆ p ) / K in terms of the known values ε p , α and K . The material-time derivative of Eq. (10.13) leads to the consistency condition for the sub-isotropic hardening surface: •

( ∂f∂(εˆεˆp ) D p ) − t r ( ∂f∂(εˆεˆp )α) = R K• + R K p

tr

p

(10.14)

Now, for the formulation of the evolution rule of the size K of the normal-isotropic hardening surface, let it be assumed that

10.2 Cyclic Stagnation of Isotropic Hardening

227

Normal-isotropic hardening surface

N

f (İ p ) = K

Dp

İp

D İˆ p Į Į

Sub-isotropic hardening surface f (İ p ) = RK

0

ε ijp

Fig. 10.5 Normal- and sub- isotropic hardening surfaces

1. The normal-isotropic hardening surface expands only if the plastic strain rate is directed outward of the sub-isotropic hardening surface, 2. The expansion rate of normal-isotropic hardening surface increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases, 3. The expansion rate of normal-isotropic hardening surface has to be zero when the plastic strain lies on its center ( R = 0) in order that the normal-isotropic • • hardening surface begins to expand gradually, noting that K = 0 for R < 0 due to the assumption 1. Based on these assumptions, let the evolution rule for magnitude of the normal-isotropic hardening surface be given by •

ˆ ˆ 〈 (∂f∂(εˆεp ) D p ) 〉 = C Rζ 〈t r (∂∂f ε(ˆεp

K = C Rζ t r

p

p)

N

)〉||D p||

(10.15)

where C ( ≤ 1) and ζ ( ≥ 1) are the material constants. Now, for the formulation of the evolution rule of the center α of the normal-isotropic hardening surface, let it be assumed that 1. It translates only if the plastic strain rate is directed outward of the sub-isotropic hardening surface,

228

10 Constitutive Equations of Metals

2. The translation rate of the center of the normal-isotropic hardening surface increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases, 3. The translation rate of normal-isotropic hardening surface has to be zero when the plastic strain lies on its center ( R = 0) in order that the normal-isotropic • hardening surface begins to translate gradually, noting that α = 0 for R < 0 due to the assumption 1 ), 4. The direction of translation is the outward-normal of the sub-isotropic hardening surface. Based on these assumptions, let the following evolution rule for the center of the normal-isotropic hardening surface be given by

α = A R ζ 〈 tr(ND p ) 〉 N = A R ζ 〈 tr(NN ) 〉 ||D p|| N

(10.16)

where

∂f (İˆ p ) ∂f (İˆ p ) N≡ || || / p ∂İˆ ∂İˆ p

(10.17)

A is the material constant which will be formulated in the following. Now, substituting Eqs. (10.15) and (10.16) into Eq. (10.14), one has t r(

• ∂f (εˆ p ) p ∂f (εˆ p ) p ∂f (εˆ p ) ζ ζ p t r D )〉 + R K − ) R R 〉 A C D t r = ( tr( 〈 R ND N 〈 ( ) ) ∂ εˆ p ∂ εˆ p ∂ εˆ p (10.18) •

Taking account of R = 0 for R = 1 in Eq. (10.18), it must hold that A = 1− C

(10.19)

Substituting Eq. (10.19) into Eq. (10.16), the evolution rule for the center of the normal-isotropic hardening surface is determined as

α = (1 − C ) R ζ 〈 tr(ND p )〉 N = (1 − C ) Rζ 〈 tr(NN) 〉 ||D p|| N

(10.20)

The normal-isotropic hardening surface expands without the translation in case of C = 1 but inversely translates without the expansion in case of C = 0 . Substituting Eqs. (10.15) and (10.20) for the evolution rules of α and K into Eq. (10.14), we have

10.2 Cyclic Stagnation of Isotropic Hardening

• R= 1 K

{¢t r ( ∂f∂(İˆİˆ

p)

p

= 1 K

{¢t r ( ∂f∂(İˆİˆ

Dp

p)

p

229

)² − t r ( ∂f∂(İˆİˆp )ĮD ) − R K•}

Dp

p

)² − t r ( ∂f∂(İˆİˆp ) (1 − C )Rζ ¢ tr(NN) ²||D p|| N) p

∂f (İˆ p ) N ∂ İˆ p

−R C R ζ ¢t r ( •

)²||D p||}

Then, R is given by

• 1 ∂f (İˆ p ) R = ¢t r ( ˆ p N )² ||D p|| [ 1 − {1 − C (1 − R )}Rζ ] K ∂İ

(10.21)

in which it holds that

⎧ 1 ∂f (εˆ p ) N ⎪= K t r ∂ εˆ p ⎪ • ⎪ p R ⎪< 1 t r ∂f (εˆ ) N ⎨ p ||D || ⎪ K ∂ εˆ p ⎪= 0 for R = 1 ⎪ ⎪⎩< 0 for R > 1

〈 (

)〉

for R = 0

〈 (

)〉

for R < 1

(10.22)

Particularly, it is noticeable that the present formulation possesses the controlling function that the plastic strain rate is always attracted to the normal-isotropic

• p R / ||D ||

1 t r ∂f (İˆ p ) K ¢ ( ∂İ p N )²

0

1

R

Fig. 10.6. Evolution of normal-isotropic hardening ratio

230

10 Constitutive Equations of Metals

hardening surface by the relation of Eq. (10.21) with Eq. (10.22) (see Fig. 10.6). In addition, the judgment whether or not the plastic strain reaches the normal-isotropic hardening surface is not required therein. Now, it is assumed that the isotropic hardening variable H evolves under the conditions: 1. The isotropic hardening is induced only if the plastic strain rate is directed outward of the sub-isotropic hardening surface, 2. The isotropic hardening rate increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases, 3. The isotropic hardening rate has to be zero when the plastic strain lies on its center ( R = 0) in order that the isotropic hardening begins to develop gradually, •

•

noting that H = 0 for R < 0 due to the assumption 1, 4. The isotropic hardening variable evolves by the rule formulated in the monotonic loading process when the plastic strain lies on the normal-isotropic hardening surface ( R = 1) . Then, let the following evolution rule of isotropic hardening be assumed by extending Eq. (6.56). •

H=

2 R υ 〈 tr(ND p )〉 = 3

2 R υ ||D p|| 〈 tr(NN) 〉 3

(10.23)

where υ ( ≥ 1) is the material constant. Accompanying with the incorporation of the cyclic stagnation of isotropic hardening, i.e. the adoption of Eqs. (10.23) instead of Eqs. (6.56), the rate of similarity-center in Eq. (8.17) with Eq. (10.6) and β = 0 for metals is given by υ + 2 aα ( rα F N − α) + 2 F' R 〈 tr(NN )〉 sˆ } (10.24) s = || D p||{c σ 3 3 R

F

Further, the plastic modulus in Eq. (8.27) is given as follows:

[ { 32 FF′R 〈tr(NN)〉 σˆ +

M p ≡ tr N

υ

2 a (r F N − α) 3 α α

}]

( ) + {U R + c( 1 − 1)}σ R R

(10.25)

On the other hand, it is insisted that the blunting of isotropic hardening in the state that the plastic strain lies inside the isotropic hardening surface is supplemented by the acceleration of kinematic hardening (Chaboche et al., 1979; Chaboche, 1989; Ohno, 1982; Ohno and Kachi, 1986). According to this notion, the plastic modulus has to be given always from Eq. (8.27) with β = 0 for metals as

10.2 Cyclic Stagnation of Isotropic Hardening

[ { 23 FF ′ σˆ +

M p ≡ tr N

231

}]

U ( R ) + c 1 − 1)} 2 a (r σ F N − α) +{ (R 3 α α R

(10.26) In order that Eq. (10.26) holds for Eq. (10.23), the evolution rule (10.6) has to be modified as follows:

α = 32 {aα (rα F N − α) + (1− Rυ 〈 tr(NN) 〉) F' σˆ}||Dp || F a ≡ 2 {aα (rα F N − α ) + (1− R υ 〈 tr(NN) 〉) F' σˆ} 3 F

(10.27)

Further, adopting Eqs. (10.23) and (10.27) instead of Eqs. (6.56) and (10.6), the evolution rule of similarity-center in Eq. (8.17) has to be modified as follows:

s = || D p||{c σ + 3 {aα (rα F N − α) + (1 − R υ )N} R 2

υ + 2 F' R 〈 tr(NN) 〉 sˆ 3 F

(10.28)

}

As described above, the two notions are considered as to the kinematic hardening rule during the blunting of isotropic hardening. The notion that the kinematic hardening is accelerated during the blunting of isotropic hardening is introduced by Ohno (1982, cf. also Ohano and Kachi, 1986; Chaboche, 1989) in order to avoid the deficiency that the isotropic hardening is restored suddenly when the plastic strain reaches the boundary of nonhardening region so that the plastic modulus change abruptly. However, this defect is always avoided predicting a smooth stress-strain curve in the present formulation based on the notion of the sub-isotropic hardening surface. Moreover, in the present formulation, the judgment whether or not the plastic strain reaches the normal-isotropic hardening surface is not required and the operation pulling back the plastic strain to the normal-isotropic hardening surface is not necessary, attracting always it to that surface. Further study is required as to whether or not the kinematic hardening is accelerated during the blunting of isotropic hardening. Because of •

• D p = Q T (Q ε pQ T )• Q = ε p − ω ε p + ε pω (ω = Q T Q )

(10.29)

232

10 Constitutive Equations of Metals

based on Eq. (4.33), the plastic strain is calculated by

ε p = ∫ (D p + ωε p − ε pω)dt

(10.30)

For the infinitesimal strain, the continuum spin W can be used as the corotational spin ω . If we adopt the Yoshida and Uemori’s (2002, 2003) idea that the isotropic hardening ceases when the back stress lies inside a certain surface, called the isotropic stagnation surface, the plastic strain ε p has only to be replaced simply by the back stress α in the formations described in this section.

10.3 On Calculation of the Normal-Yield Ratio The normal-yield ratio R can be calculated directly by R = f (σˆ )/F in the initial subloading surface model. However, it has to be calculated by solving the equation of the subloading surface in the extended subloading surface model as described below. Substituting Eq. (10.2) into Eq. (8.9) with β = 0 , the extended subloading surface is described as follows:

3 ||σ' + R sˆ' ||= RF ( H ) 2

(10.31)

i.e. tr(ı ' + Rsˆ' ) 2 = 2 R 2 F 2 3

(10.32)

The normal-yield ratio R is derived from the quadratic equation (10.32) as follows: tr(σ' sˆ') + R=

tr 2 (σ' sˆ' ) + ( 2 F 2 − || sˆ' ||2 ) ||σ' || 2 3 2 F 2 − ||sˆ || 2 ' 3

(10.33)

10.4 Comparisons of Test Results Capability of the extended subloading surface model for the prediction of deformation behavior of metals is verified by comparisons with some basic test data below (Hashiguchi and Ozaki, 2009a,b).

10.4 Comparisons of Test Results

233

The cyclic loading behavior under the stress amplitude in both positive and negative sides can be predicted to some extent by any models including even the conventional plasticity model. On the other hand, the prediction of the cyclic loading behavior under the stress amplitude in positive or negative one side, i.e. the pulsating loading inducing the so-called mechanical ratcheting effect requires a high ability for the description of plastic strain rate induced by the rate of stress inside the yield surface. Further note that we quite often encounters the pulsating loading phenomena in the boundary-value problems in engineering practice, e.g. railways, gears and plates undergoing punch-indentation. The comparison with the test data of the uniaxial deformation behavior of 1070 steel under the pulsating loading between 0MPa and 830Mpa after Jiang and Zhang (2008) is depicted in Fig. 10.7 where the material constants and the initial values are selected as follows: Material constants:

Elastic moduli: E = 170000MPa, ν = 0.3, ⎧isotropic : h1 = 0.4, h 2 = 170, Hardening ⎨ ⎩kinematic : aα = 100, rα = 0.3, Evolution of normal - yield ratio : Re = 0.1, u 0 = 500, us = 2, Translationon of similarity - center : c = 200, Stagnation of isotropic hardening : C = 0.5, ζ = 20, υ = 3, Initial values:

Hardening function: F0 = 580MPa, Size of the normal-isotropic hardening surface : K 0 = 0.001, Kinematic hardening variable : Į 0 = 0 kPa, Center of similarity : s0 = 0MPa , Stress : ı 0 = 0MPa The test result is simulated very closely by the present model. The movements of kinematic hardening variable α and the similarity-center s are shown in that figure where the axial components are designated by ( )a . In addition, the variations of normal-yield ratio R and the normal-isotropic hardening ratio R are depicted. It is shown that the normal-yield ratio R does not increase over unity and thus the stress is automatically attracted to the normal-yield surface without

234

10 Constitutive Equations of Metals

σa Experiment Prediction 800 600

sa

σ a (MPa) 400 200

αa 0.005

0.01

0.015

εa

0

0.005

0.01

0.015

εa

0

0.005

0.01

0.015

εa

0

1.0 R

0.5

1.0 R

0.5

Fig. 10.7 Prediction of zero-to-tension uniaxial loading behavior of 1070 steel (test data after (Jian and Zhang, 2007))

10.4 Comparisons of Test Results

235

σa Experiment Prediction

800 600

σ a (MPa) 400 200

αa 0

0.005

0.01

0.015

εa

(a) c = 0 : Initial subloading surface model

σa Experiment Prediction

800

sa

600

σ a (MPa) 400 200

αa 0

0.005

0.01

0.015

εa

(b) us = 0 (u = const.)

Fig. 10.8 Prediction of zero-to-tension uniaxial loading behavior of 1070 steel (test data after (Jian and Zhang, 2007)): (a) Initial subloading surface model, (b) u=const.

incorporation of any return-mapping algorithm. Also, the normal-isotropic hardening ratio R does not increase over unity and thus the normal-isotropic hardening surface changes such that the plastic strain does not go out from this surface.

236

10 Constitutive Equations of Metals

The simulation by the initial subloading surface model in which the center of similarity s is fixed in the origin of stress space by setting c = 0 is shown in Fig. 10.8(a), while the other parameters are chosen same as those for Fig. 10.7. It is known that the subloading surface contracts and then the unloading proceeds during the decrease of stress to zero so that open hysteresis loopa are depicted predicting an unrealistically large strain accumulation. Further, the simulation in the case that the material parameter u in the evolution rule of the normal-yield ratio is taken to be constant is shown in Fig. 10.8(b). It is shown that the excessively large plastic strain is induced in the reloading processes as was described in 8.4 and thus the unrealistically large strain accumulation is predicted even in this calculation although the strain accumulation is much improved to be suppressed from the calculation by the initial subloading model in Fig. 10.8(a). As seen in numerical experiments shown in Appendix 7, the strain accumulation is suppressed for the deformation behavior near the yield state by the extension of the material parameter u to Eq. (8.36). The comparison with test data for the cyclic loading behavior under the increasing strain amplitudes ±1, ± 1.5, ± 2, ± 2.5, ± 3% in turn after saturation in each amplitude is shown in Fig. 10.9. The test data are obtained for the 316L steel at 20 C after Chaboche et al. (1979). The material constants and the initial values are selected as follows: Material constants:

Elastic moduli: E = 170000MPa, ν = 0.3, ⎧isotropic : h1 = 0.8, h 2 = 10, Hardening ⎨ ⎩kinematic : aα = 100, rα = 0.4, Evolution of normal - yield ratio : u 0 = 800, us = 3, Re = 0.4, Translationon of similarity - center : c = 200, Stagnation of isotropic hardening : C = 0.5, ζ = 15, υ = 3, Initial values:

Hardening function: F0 = 275 MPa, Size of the normal-isotropic hardening surface : K 0 = 0.001, Kinematic hardening variable : Į 0 = 0MPa , Center of similarity : s0 = 0MPa , Stress : ı 0 = 0MPa This test result is also predicted very closely. The variations of normal-yield ratio R and the normal-isotropic hardening ratio R do not increase over unity as shown in Fig. 10.10.

10.4 Comparisons of Test Results

237

(MPa) σσ (MPa)

σ (MPa) 500

250

αa

(%) εεaa (%)

−3

−2

0

−1

1

2

3

ε (%)

−250

−500

(b) Prediction

(a) Experiment

Fig. 10.9 Comparison with experiment for the cyclic loading behavior under the five levels of constant strain amplitudes in both positive and negative sides of 316L steel (test data after Chaboche et al.,1979)

R− 1.0 1.0

R 1.0 1.0

0.5 0.5

0.5 0.5

− 0.3 − 0.2 − 0.1 0.0 0.1

Fig. 10.10 Variations of

R

0.2 0.3

−− 0.3 − 0.2 − 0.1 0.0 0.1

0.2 0.3

and R in calculation for Fig. 10.8

The prediction without the stagnation of isotropic hardening by setting C = 0 or K = 0 is shown in Fig. 10.11. The isotropic hardening develops unrealistically rapidly in this calculation.

238

10 Constitutive Equations of Metals

σa (MPa) 500

250

αa −3

−2

−1

0

1

2

3

ε a (%)

−250

−500

Fig. 10.11 Prediction of the cyclic loading behavior under the five levels of constant strain amplitudes in both positive and negative sides of 316 L steel (test data after Chaboche et al.,1979) without stagnation of isotropic hardening

10.5 Orthotropic Anisotropy The kinematic hardening incorporated in the foregoing is regarded to be the induced anisotropy. On the other hand, various inherent anisotropies are induced in the manufacturing process of metals. The typical inherent anisotropy is the orthotropic anisotropy formulated by Hill (1948). Now, consider the general yield function in the quadratic form shown as follows:

f (σ ij ) = 1 Cijklσ ijσ kl 2

(10.34)

where Cijkl is the fourth-order anisotropic tensor having eighty-one components fulfilling

Cijkl = Cijlk = C jikl = C jilk = Cklij = Cklji = Clkij = Clkji

(10.35)

by the relations, Cijklσ ij σkl (= Cklijσkl σ ij ) = Cklijσ ijσkl based on the symmetry of the stress tensor σ ij = σ ji . Then, the independent components reduces to twenty-one leading to

10.5 Orthotropic Anisotropy

239

Cijklσ ijσ kl = C1111σ 112 + 2C1122σ 11σ 22 + 2C1133σ 11σ 33 + 2C1112σ 11σ 12 + 2C1123σ 11σ 23 + 2C1131σ 11σ 31 2 + C2222σ 22

+ 2C2233σ 22σ 33 + 2C2212σ 22σ 12 + 2C2223σ 22σ 23 + 2C2231σ 22σ 31 + C3333σ 332

+ 2C3312σ 33σ 12 + 2C3323σ 33σ 23 + 2C3331σ 33σ 31 + C1212σ 122

+ 2C1223σ 12σ 23 + 2C1231σ 12σ 31 2 + C2323σ 23

+ 2C2331σ 23σ 31 + C3131σ 312 )

(10.36) which is the general form of yield function in the quadratic form. Here, assuming the plastic incompressibility, it holds that

(∂ (2 f ) / ∂σ pq )δ pq = (∂ Cijklσ ijσ kl /∂σ pq )δ pq = Cijklδ pi δ qjσ klδ pq + Cijklσij δ pk δ qlδ pq = C pp klσ kl + Cij ppσij = C pp klσ kl + C ppijσij = 2C ppklσ kl = 0 This relation must hold for any σij and thus one obtains (10.37)

C ppkl = Cij qq = 0

which leads to

C1111 + C1122 + C1133 = 0 ⎫ C2211 + C2222 + C2233 = 0 ⎪⎬ C3311 + C3322 + C3333 = 0 ⎪⎭

(10.38)

C1112 + C2212 + C3312 = 0 ⎫ C3312 = −(C1112 + C2212 ) ⎫ C1123 + C2223 + C3323 = 0 ⎪⎬ → C1123 = −(C2223 + C3323 ) ⎬⎪ C1131 + C2231 + C3331 = 0 ⎪⎭ C2231 = −(C1131 + C3331 ) ⎭⎪

(10.39)

The substitution of Eq. (10.39) into Eq. (10.36) gives the expression Cijklσ ijσ kl = C1111σ 112 + 2C1122σ 11σ 22 + 2C1133σ 11σ 33 2 + 2C2233σ 22σ 33 + C2222σ 22

+ C3333σ 332

+ 2C1112σ 11σ 12 − 2(C2223 + C3323 )σ 11σ 23 + 2C1131σ 11σ 31 + 2C2212σ 22σ 12

+ 2C2223σ 22σ 23 − 2(C1131 + C3331 )σ 22σ 31

− 2(C1112 + C2212 )σ 33σ 12 + 2C3323σ 33σ 23 + C1212σ

2 12

+ 2C1223σ 12σ 23 2 + C2323σ 23

+ 2C3331σ 33σ 31 + 2C1231σ 12σ 31 + 2C2331σ 23σ 31 + C3131σ 312

(10.40)

240

10 Constitutive Equations of Metals

Further, considering Eq. (10.38), one has 2 + C3333σ 332 + 2C1122σ 11σ 22 + 2C 2233σ 22σ 33 + 2C1133σ 11σ 33 C1111σ 112 + C 2222σ 22

2 + C3333σ 332 = C1111σ 112 + C2222σ 22

2 − C1122 (σ11 − σ 22 )2 + C1122σ112 + C1122σ 22 2 − C2233 (σ 22 − σ 33 ) 2 + C2233σ 22 + C2233σ 332

− C1133 (σ 33 − σ 11 ) 2 + C1133σ 332 + C2233σ 112 2 + (C1133 + C2233 + C3333 )σ 332 = (C1111 + C1122 + C2233 )σ 112 + (C1122 + C2222 + C2233 )σ 22

− C1122 (σ 11 − σ 22 ) 2 − C 2233 (σ 22 − σ 33 ) 2 − C1133 (σ 33 − σ 11 ) 2

= − C1122 (σ 11 − σ 22 ) 2 − C2233 (σ 22 − σ 33 ) 2 − C1133 (σ 33 − σ 11 ) 2

(10.41)

Then, by setting

a1 ≡ − C1122 , a2 ≡ − C2233 a3 ≡ − C1133 , a4 ≡ − 2C1112 , a5 ≡ − 2C2212 , a6 ≡ − C2223 , a7 ≡ − 2C3323 a8 ≡ − 2C3331 , a9 ≡ − 2C1131 , a10 ≡ 2C1223 , a11 ≡ 2C2331 a12 ≡ 2C1231 , a13 ≡ C1212 , a14 ≡ C2323 a15 ≡ C3131 Eq. (10.40) is rewritten as

Cijklσ ijσ kl = a1 (σ 11 − σ 22 ) 2 + a2 (σ 22 − σ33 ) 2 + a3 (σ 33 − σ 11 )2

+{a4 (σ 33 − σ 11 ) + a5 (σ 33 − σ 22 )}σ 12 +{a6 (σ 11 − σ 22 ) + a7 (σ 22 − σ 33 )}σ 23

+{a8 (σ 22 − σ 33 ) + a9 (σ 22 − σ 11 )}σ 31 + a10σ 12σ 23 + a11σ 23σ 31 + a12σ 31σ 12 2 + a15σ 312 + a13σ 122 + a14σ 23

(10.42)

Equation (10.42) is the general yield function for the plastically-incompressible materials in the quadratic form. Furthermore, assume orthotropic anisotropy. Then, if we describe the yield surface by the coordinate axes selected to the principal axes {e∗i } of orthotropic

10.5 Orthotropic Anisotropy

241

anisotropy, the yield function is independent of the sign of shear stress components in this coordinate system. Therefore, it must hold that

a4 = a5 = a6 = a7 = a8 = a9 = a10 = a11 = a12 = 0 Here, replacing the symbols

a i as

F = a1 , G = a2 , H = a3 , L = a13 / 2, M = a14 / 2, H = a15 / 2 used by Hill (1948), Eq. (10.42) leads to the Hill’s yield condition with orthotopic anisotropy: f (σ ij ) =

1 (σ − σ ) 2 2 {F 11 + G (σ 22 − σ 33 ) 2 + H (σ 33 − σ 11 ) 2 + 2 Lσ 122 + 2 M σ 23 + 2 Nσ 312 } 22 2

(10.43)

= F (H )

Here, note that F = G = H = 1, L = M = N = 3 holds for isotropy and then Eq. (10.43) reduces to f (σ ij ) = 3/2 ||σ' || which is the equivalent stress. While Eq. (10.43) is the expression on the principal axis {e∗i } of orthotropic anisotropy, it is rewritten by the following equation stipulating this fact.

f (σ ij∗) =

1 (σ ∗ − σ ∗ ) 2 {F 11 + G (σ 22∗ − σ 33∗ )2 + H (σ 33∗ − σ 11∗) 2 + 2 Lσ 12∗2 + 2M σ 23∗ 2 + 2 Nσ 31∗2 } 22 2

(10.44)

= F (H )

Further, under the plane stress condition observed in the sheet metal forming it holds that σ 23∗ = σ 31∗ = σ 33∗ = 0 and thus Eq. (10.44) reduces to

( H + F )σ 11∗ 2 − 2 Fσ 11∗σ 22∗ + (G + F )σ 22∗ 2 + 2 Lσ 12∗2 = 1

(10.45)

where

H≡

G F H L , G≡ , F≡ ,L≡ 2{F ( H )}2 2{F ( H )}2 2{F ( H )}2 2{F ( H )}2 (10.46)

Here, denoting the yielding strength in the equi-two axis tension as the pure shear as τ , it holds from Eq. (10.45) that

σ and that of

242

10 Constitutive Equations of Metals

σ ≡ ( H + G ) −1/2 , τ ≡ (2L ) −1/2

(10.47)

Now, rewrite Eq. (10.45) as { 1 (G + H ) + 1 (G + H + 4 F ) − 1 (G − H )}σ 11∗ 2 + { 2 (G + H ) − 2 (G + H + 4 F )}σ 11∗σ 22∗ 4 4 2 4 4 + { 1 (G + H ) + 1 (G + H + 4 F ) + 1 (G − H )}σ 22∗ 2 + 2 Lσ 12∗2 = 1 4 4 2

which is arranged as follows:

1 (G + H )(σ ∗ + σ ∗ )2 + 1 (G + H + 4 F )(σ ∗ − σ ∗ )2 22 22 11 11 4 4 − 1 (G − H )(σ 11∗2 − σ 22∗ 2 ) + 2 Lσ 12∗2 = 1 2

(10.48)

Denoting the angle measured in the counterclockwise direction from the principal axes of anisotropy to the principal stress as

α

and substituting the relations

σ 11∗ + σ 22∗ = σⅠ + σⅡ, σ 11∗ − σ 22∗ = (σⅠ − σⅡ) cos 2α , 2σ 12∗ = (σⅠ − σⅡ) sin 2α

(10.49)

into Eq. (10.48), one has

1 (G + H )(σ + σ )2 + 1 (G + H + 4 F )(σ − σ )2 cos2 2α Ⅰ Ⅱ Ⅰ Ⅱ 4 4 − 1 (G − H )(σⅠ2 − σⅡ2 ) cos 2α + 1 L (σⅠ − σⅡ)2 sin 2 2α = 1 2 2 which is rewritten as

(σⅠ + σⅡ)2 − 2a (σⅠ2 − σⅡ2 ) cos 2α + b(σⅠ − σⅡ)2 cos2 2 +2

L (σⅠ − σⅡ)2 = 4 G+H G+H

(10.50)

where

a ≡ G − H , b ≡ G + H + 4F − 2L G+H G+H The substitution of Eq. (10.47) into Eq. (10.50) leads to 2 σ σ 2 σ2 σ2 (σⅠ + σⅡ )2 + (σ τ ) ( Ⅰ − Ⅱ) − 2a( Ⅰ − Ⅱ) cos 2α +

(10.51)

10.5 Orthotropic Anisotropy

243

α + b(σ − σ )2 cos2 2α = (2σ ) 2

(10.52)

Equation (10.52) is extended to the following equation for the plane isotropy with the material constant m ( ≥ 1) . m σ m σ m |σⅠ + σⅡ| m +(σ τ ) | Ⅰ − Ⅱ| = (2σ )

(10.53)

Hill (1990) proposed the following extended orthotropic yield condition from Eqs. (10.52) and (10.53). m σ σ m |σⅠ + σⅡ| m +(σ τ ) | Ⅰ − Ⅱ|

+ |σⅠ2 + σⅡ2 | (m /2) −1 {−2 a (σⅠ2 − σⅡ2 ) + b (σⅠ − σⅡ )2 cos 2α }cos 2α = (2σ ) m (10.54)

Equation (10.54) involves the five material constants, i.e. the yield stress σ , τ and the dimensionless number a, b, m . It reduces to Eq. (10.52) for m = 2 and to Eq. (10.53) for a = b = 0 (or α = π / 4 ). By use of Eq. (10.49), Eq. (10.54) is rewritten in the anisotropic axes as follows: ∗ | m + (σ ) m |(σ 11∗ − σ 22 ∗ )2 + 4σ 12∗ 2 | m /2 |σ 11∗ + σ 22 τ + |(σ 11∗ + σ 22∗ )2 + 4σ 12∗2 | (m /2) −1 {−2a(σ 11∗2 − σ 22∗ 2 ) + b(σ 11∗ − σ 22∗ )2 } = (2σ ) m

(10.55) Generally, the yield surface is described in the principal axes of anisotropy as follows:

f (σ ij∗) = F ( H )

(10.56)

σ ij∗ = Qir Q jsσrs

(10.57)

where

Qij (t ) ≡ ei∗(t ) • e j (= cos(e ∗i (t ), e j ))

(10.58)

Needless to say, Eq. (10.56) is not a general tensor expression but is merely the expression by the components. The variation of e∗i is calculated using the following equation with the initial value of e∗i 0 .

e∗i = e∗i 0 + ∫ e• ∗i dt • where e∗i is given by

(10.59)

244

10 Constitutive Equations of Metals

e• ∗i = ωe∗i

(10.60)

denoting the spin of principal axes of orthotropic anisotropy as ω . Here, the stress rate σ• ij∗ is calculated by

σ• ij∗ = σ ij∗ = Q ir Q js (σ• rs − ωrpσ ps + σ rpω ps)

(10.61)

for the case of the input of stress rate σ• rs from Eq. (4.33) and by

σ• ij∗ = Q ir Q js σ rs

(10.62)

after calculating the corotational stress rate σ rs from the constitutive relation for the case of the input of the strain rate.

10.6 Representation of Isotropic Mises Yield Condition The isotropic yield function described by Eq. (6.54) can be expressed in the following various forms. f (σ) = σ e = 3

=

Ⅱ'

σ

=

3 || σ || = 2 '

3 σ σ 2 'rs 'rs

3 σ ' + σ ' + σ ' + 2(σ ' + σ ' + σ ' ) 112 222 332 12 232 312 2

= 1 {(σ 11 − σ 22 )2 + (σ 22 − σ 33 )2 + (σ 33 − σ 11 )2 + 6(σ 122 + σ 232 + σ 312 )}1/ 2 2

=

3 σ' + σ' + σ' 2 Ⅱ2 Ⅲ2 2 Ⅰ

= 1 (σⅠ − σⅡ)2 + (σⅡ − σ Ⅲ )2 + (σ Ⅲ − σⅠ)2 = F 2

(10.63)

The combined test of the tensile stress σ ( = σ11 ) and the distortional stress τ ( = σ 12 ) for a thin wall cylinder specimen is widely adopted for metal. In this case Eq. (10.63) is rewritten as

σ 2 + ( 3τ ) 2 = F 2

(10.64)

10.6 Representation of Isotropic Mises Yield Condition

245

Then, the Mises yield condition is shown by a circle of radius F in the (σ , 3τ ) plane. The visualization of the stress state can be realized in the space of three and less dimension. The stress state can be represented completely in the principal stress space when principal stress directions are fixed to materials and only the principal stress values change. In general, however, one must use the six-dimensional space or memorize the variation of the principal stress direction if the directions change. However, in the cases for which the number of independent variable components is less than three, such as the tension-distortion test described above and the plane stress and strain tests, the state of stress can be represented in the three and less dimensional stress space. The Ilyushin’s isotropic stress space (Ilyushin, 1963) is convenient to depict the Mises yield surface, which depends only on the deviatoric stress, as explained below. The deviatoric stress tensor involves the five independent variables and thus the Mises yield surface in Eq. (10.63) is described by the independent components as follows:

f (σ ) = 3σ '112 + 3σ '222 + 3σ11 ' σ '22 + 3(σ '12 + σ '232 + σ '312 ) =

( 23 σ' )

2

11

+ 3( 1 σ 11 ' + σ'22 ) + 3(σ12' + σ '232 + σ'312 ) = F 2 2

and thus it can be rewritten as

S12 + S 22 + S32 + S 42 + S52 = F 2

(10.65)

in the five-dimensional space with the axes

(

S1 = 3 σ11 + σ '22 , S3 = 3σ12 , S 2 = 3 1 σ11 ' , S4 = 3σ '23 , S5 = 3σ '31 2 ' 2 '

)

(10.66) Equation (10.65) exhibits the five-dimensional super-spherical surface. Further, consider the expression of the Mises yield surface for the plane stress and strain conditions in the following. 10.6.1 Plane Stress State

The plane stress state fulfilling σ 3 j = 0 can be described in the three-dimensional space (σ 11 , σ 22 , σ 12 ) and thus the Mises yield condition (10.63) is described by the following equation.

σ 112 − σ 11σ 22 + σ 222 + 3σ 122 = F

(10.67)

246

10 Constitutive Equations of Metals

On the other hand, Eq. (10.67) can be described in the two-dimensional principal stress plane as follows:

σⅠ2 − σⅠσⅡ + σⅡ2 = F

(10.68)

which is the section of the Mises yield condition cut by the plane σ Ⅲ = 0 and exhibits Mises’s ellipse in the principal stress plane (σⅠ, σⅡ) as shown in Fig. 10.12. It holds from Eq. (1.219) 3 because of σ Ⅲ = 0 leading to σ m = −σ Ⅲ' that

σ m = − 2 F cos (θ + 2 π )

(10.69)

3

3

Substituting Eq. (10.69) into Eq. (1.219), one obtains ⎫ σⅠ = − 2 F cos (θ + 2 π ) + 2 Fcos θ = 2 F sin (θ + π ) ⎪ 3

3

3

3

σⅡ = − 2 F cos (θ + 2 π ) + 2 F cos (θ − 2 π ) 3

3

3

3

3

⎪ ⎬ = 2 F sinθ ⎪ 3 ⎪⎭

(10.70)

from which the coordinates of main points on the Mises’s ellipse are calculated as shown in Fig. 10.11. The thin curve shows the Hill’s orthotropic Mises yield surface in Eq. (10.45), which is rotated the principal axes of ellipse with the changes of its long and short radii from the isotropic Mises yield surface. Next, consider the Ilyushin’s isotropic stress space in which the variables in Eq. (10.66) are used. Here, in the present case fulfilling σ 3 j = 0 leading to S4 = S5 = 0 the Mises yield surface is represented by the sphere in the ( S1 , S2 , S3 ) space, while it holds that

⎫ ⎪ ⎪ S2 = 3{(1/2)σ 11 ' + σ '22 ) ⎬ ⎪ = 3[(1/2){σ 11 − (σ 11 + σ 22 ) / 3} + {σ 22 − (σ 11 + σ 22 ) / 3}] = ( 3/2)σ 22 ⎪ ⎭ S1 = (3/2)σ 11 ' = (3/2){σ 11 − (σ 11 + σ 22 ) / 3} = σ 11 − σ 22 / 2

(10.71) Furthermore, in the case fulfilling σ 12 = 0 , the Mises yield surface is represented by the circle in the ( S1 , S 2 ) plane (Fig. 10.13). Here, setting

10.6 Representation of Isotropic Mises Yield Condition

σ

(

1 , F 3

(θ =120D ) F

(−

247

2 F ) (θ = 90D ) 3 ( F , F ) (θ = 60D )

1 F , 1 ) (θ =150D ) F 3 3

( 2 F, 3

1 ) (θ = 30D ) F 3

(θ =180D ) − F

F (θ = 0)

0 (θ = 210D ) (− 23 F , − 1 F ) 3

(1

3 D D − F (θ = 300 ) (θ = 240 ) (− F , − F ) 2 D 1 (− 3 F , − 3 F ) (θ = 270 )

σ

F , − 1 F ) (θ = 330D ) 3

Fig. 10.12 Mises yield surface in the plane stress condition. (Thin curve describes Hill’s orthotropic Mises yield condition).

S2

F

φ

−F 0

F S1

−F Fig. 10.13 Mises yield surface in plane stress state without shear stress (σ 12 =0)

S1 = Fcosφ , S 2 = Fsinφ

(10.72)

and substituting them into Eq. (10.71), it holds that

σ11 = 2 F sin(φ + π ), σ 22 = 2 F sin φ 3

3

3

(10.73)

248

10 Constitutive Equations of Metals

10.6.2 Plane Strain State

If the elastic strain rate can be ignored compared with the plastic strain rate in the plane strain state, the following relation holds by substituting D33p = λσ 33' = 0 into σ rr' = 0 .

σ 33 = 1 (σ 11 + σ 22 ) 2

(10.74)

Then, the Mises yield surface is described from Eq. (10.63) 3 by 3

(σ

11

− σ 22 2 ) + σ122 = F 2

(10.75)

which is represented by the Mohr’s circle in the plane of the normal and the shear stresses.

Chapter 11

Constitutive Equations of Soils

11 Constitutive Equatio ns of So ils

The history of plasticity has begun with the study on deformation behavior of soils by Coulomb (1773) when he has proposed the yield condition of soils by applying the friction law of himself. Thereafter, the soil plasticity has been deprived the leading part by the metal plasticity. One of the reasons would be caused by the fact that soils exhibit various complex plastic deformation behavior, e.g., the pressure-dependence, the plastic compressibility, the dependence of the third invariant of deviatoric stress, the softening and the rotational hardening. Explicit constitutive equations of soils will be described in this chapter, based on the elastoplastic constitutive equations in Chapters 6-8.

11.1 Isotropic Consolidation Characteristics The ln v − ln p linear relation ( p ≡ −(t r σ)/3 : pressure) for the isotropic consolidation characteristics of soils was proposed by Hashiguchi (1974; see also 1985, 1995, 2008; Hashiguchi and Ueno, 1977), which is depicted in Fig. 11.1. py + p ⎫ vy − ρ ln p + pe ⎪ = vy 0 y0 e⎪ ⎬ p + pe ⎪ v Swelling line: ln = −γ ln p + p ⎪ e 0 V ⎭ Normally - consolidated line: ln

(11.1)

where ( p0 , V ) , ( p y 0 , Vy 0 ) , ( p y , Vy ) are the pressures and the volumes in the initial state, the initial yield state and the current yield state, respectively. V is the volume in the unloaded state to the initial pressure, corresponding to the intermediate configuration described in 6.1. Further, ρ and γ are the material constants prescribing the slopes of normal-consolidation and swelling lines, respectively, in the (ln v, ln p) plane. pe (>0) is the material constant prescribing the pressure for which the volume becomes infinite, i.e. v → ∞ for p → − pe. K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 249–307. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

250

11 Constitutive Equations of Soils

ln v

Swelling line

V vy0

Normal-consolidation line

1 ρ V

Swelling line

1 γ

v vy

0

p0 + pe

py0 + pe

p + pe

py + pe

ln( p + pe )

Fig. 11.1 ln v − ln p linear relation of isotropic consolidation of soils

The logarithmic strain and its elastic and plastic components are given from Eq. (6.7) and (11.1) as follows:

ε v = ε ve + ε vp = ln v + ln V = ln v + (ln vy 0 + ln vy + ln V ) V vy vy V V V 0

p y 0 + pe p +p py + p p+ p = −γ ln p + pe + ( −γ ln p + p − ρ ln p + pe − γ ln p0 + pe ) y e 0 y0 e e e 0 py + p p+ p = −γ ln p + pe − ( ρ − γ ) ln p + pe y0 e e 0

(11.2)

i.e.

p p ε v = −γ ln p + pe − ( ρ − γ ) ln y + e , ⎪⎫ p0 + pe p y 0 + pe

⎪ ⎪⎪ (11.3) ⎬ ⎪ p y + pe ⎪ p ε v = −( ρ − γ ) ln p + p ⎪ y0 e ⎪⎭ The volumetric strain rate and its elastic and plastic components are given from Eq. (11.3) as follows: p p ε ve = −γ ln p + pe , 0 + e

11.1 Isotropic Consolidation Characteristics

251

p• y p• Dv = −γ p p − ( ρ − γ ) p p + e y+ e p• Dve = −γ p p + e

Dvp = − ( ρ − γ ) p

p• y y + pe

⎫ ⎪ ⎪⎪ ⎬ ⎪ ⎪ ⎪⎭

(11.4)

which show the volumetric elastic and the plastic strain rates derived exactly from the ln v − ln p linear relation based on the multiplicative decomposition of deformation gradient. Strictly speaking, the elastic deformation characteristic are influenced by plastic deformation in general, but p y is not incorporated in the elastic strain and its rate in Eqs. (11.3) and (11.4) in order to avoid the elastic-plastic coupling, which would raise the complexity of formulation, whilst the coupling in soils is not so strong as in brittle geomaterials, e.g. rocks and concretes. Adopting Eq. (11.4) in the explicit equation (5.28) for the elastic modulus tensor E in Eq. (6.29), the elastic bulk modulus K and the elastic shear modulus G are given as

K=

p + pe

γ

,

G=

3(1 − 2ν ) K 2(1 + ν )

(11.5)

On the other hand, the e − ln p linear relation for the isotropic consolidation has been widely adopted for constitutive equation of soils including Cam-clay models (Roscoe and Burland, 1968; Schofield and Wroth, 1968) in which the normal-consolidation and the swelling lines are given by

py ⎫ Normally - consolidated line: e y − ey 0 = −λ ln p ⎪ y0 ⎪ ⎬ p ⎪ Swelling line: e − e = −κ ln p ⎪ 0 ⎭

(11.6)

where the material constants λ and κ are the slopes of normal-consolidation and swelling lines, respectively, in the ( e, ln p ) plane (Fig. 11.2), where ( p0 , e 0 ) , ( p y 0 , e y 0 ) and ( p y , e y ) are the pressure, the void ratio in the initial, the initial yield and the current yield states, respectively. In addition, e is the void ratio in the unloaded state to the initial pressure p0 .

252

11 Constitutive Equations of Soils

ln e

Swelling line

e0 ey0

Normal-consolidation line 1

λ

e 1

e ey

0

p0

py0

p

Swelling line

κ

py

ln p

Fig. 11.2 e − ln p linear relation for isotropic consolidation of soils

However, the e − ln p linear relation has the following physical impertinence. 1.

2. 3.

4. 5.

The change of void ratio induced during the same range of pressure along the swelling line is independent of a pre-consolidation pressure py , although the plastic decrease of void ratio proceeds with the increase of py . The void ratio becomes negative if the pressure becomes large as py > py 0 exp( ey 0 / λ ) or p > p0 exp(e / λ ) . The relation is not given by the volume but by the void ratio. Therefore, the relation of pressure vs. volumetric strain becomes quite complicated even if the incompressibility of soil particles is assumed. The volume becomes infinite when the pressure approaches zero. The relation is given by the difference of void ratio. Therefore, it fits to the nominal strain but becomes quite complex for the logarithmic strain which is pertinent for finite strain.

Therefore, the following nominal volumetric strain has been adopted in constitutive equations.

ε v = v − V , ε ve = v − V , ε vp = V − V V V V

(11.7)

Here, it is noteworthy that the nominal strain cannot be related to the strain rate D in the exact sense. The following approximation is used by substituting the void

ratio instead of the volume into Eq. (11.7) in order to evade the above-mentioned deficiency 3.

11.2 Yield Conditions

253

ε v ≅ e − e0 , ε ve ≅ e − e , ε vp ≅ e − e0 = 1 + e0 1 + e0 1 + e0

( ey 0 − e0 ) + (ey − ey 0) + (e − ey ) 1 + e0 (11.8)

Substituting Eq. (11.6) into Eq. (11.8), one has p ε v ≅ − κ ln pp − λ − κ ln p y , ε ve ≅ − κ ln p p0 1 + e0 1 + e0 y0 1 + e0 0

p p p p ε vp ≅ − κ ln py 0 − λ ln p y − κ ln p 0 = − λ − κ ln p y (11.9) e e e + + + 1 + e0 1 1 1 y0 y y0 0 0 0 0 from which the nominal volumetric strain rate is given by • p• λ − κ py , − 1 + e0 p 1 + e0 py

• εv ( ≅ Dv ) ≅ − κ

• • p p• • − p ε ve (≅ Dve ) ≅ − κ e p , ε vp ( ≅ Dv ) ≅ − λ eκ py 1+ 0 1+ 0 y

(11.10)

Equation (11.10) is adopted most widely for elastoplastic constitutive equations of soils. Nevertheless, it has the following deficiencies. 6. 7.

It is derived merely approximately from the e − ln p linear relation. It cannot be adopted to describe finite deformation since it is derived based on the definition of nominal strain, which can be adopted only for the description of an infinitesimal deformation.

8.

The tangent elastic bulk modulus K ( ≡ p• / ε•ve ) is given by K = (1 + e0 ) p / κ from Eq. (11.10). It has a crucial physical impertinence: “The larger the void ratio, the larger is the elastic bulk modulus, i.e., the looser the soil, the smaller is the change of void ratio to be induced”. In addition, the tangent elastic bulk modulus K depends on the initial void ratio e0 . Consequently, it is not a material constant.

Eventually, it is concluded that the e − ln p linear relation is inadequate physically and mathematically for formulation of constitutive equations for finite deformation of soils. On the other hand, the various deficiencies in the e − ln p linear relation are remedied completely in the ln v − ln p linear relation.

11.2 Yield Conditions Various yield conditions of soils have been proposed to date. The functions f (σ ) in Eq. (6.30) can be reduced to the unique form of

254

11 Constitutive Equations of Soils

f (σ) = pg ( χ )

(11.11)

where

χ ≡ ||η || M

， η ≡ σp'

(11.12)

M is the stress ratio ||η || in the maximum point of ||σ' || , i.e. the critical state and is called the critical state stress ratio. The following explicit equation of M was proposed (Hashiguchi, 2002). M ( cos3θ σ ) =

14 6 sin φc (3 − sin φc )(8 + cos3θ σ )

(11.13)

where cos 3θ σ ≡ 6t r τ 3 , τ ≡ σ'

||σ' ||

(11.14)

φ c is the angle of internal friction in the critical state for the axisymmetric compression stress state, i.e. the so-called tri-axial compression (θ σ = π / 3) . The section of the conical surface ||η || = M cut by the π plane is depicted in Fig. 11.3. It fulfills the convexity condition (cf. Eq. (A.14) in Appendix 5) 1 + d 1 2 8 3 − sin φc M dθ σ2 ( M ) = 14 6 sin φ (1 − cos3θ σ ) ≥ 0 c

(11.15)

for the whole range of φ c . On the other hand, the following simple equation has been widely used for constitutive equations of soils (Satake, 972; Gudehus, 1973; Argyris et al., 1973).

M (θ σ ) =

2 6 sin φc 3 − sinφc sin 3θ σ

(11.16)

However, Eq. (11.16) violates the convexity condition

1 1 2 1 d sin φcsin 3θ σ ) < 0 M + dθ σ2 ( M ) = 2 6 sin φ (3 + 8 c for φ c > 22 01' .

(11.17)

11.2 Yield Conditions

255

σ3 φc = 45D

30D 15D

0

σ2

σ1 θı

Fig. 11.3 Section of the critical state surface of soils by π–plane (Hashiguchi, 2002)

It is premised in Eq. (11.11) that the yield surface passes through the isotropic compression state and the null stress point. Then, the function g ( χ ) must fulfill the following conditions g ( χ ) = 1 for χ = 0 ⎫ g ( χ ) → ∞ for χ → ∞ ⎬⎭

(11.18)

Furthermore, it is premised that the magnitude of the deviatoric stress, ||σ' || , becomes maximal at the critical state and thus it holds that

g' (1) = g (1)

(11.19)

(cf. Appendix 6). Denoting the p -value in the critical state as pcr , the following equation is obtained by substituting Eq. (11.11) into Eq. (6.30) at χ = 1 .

pcr = F / g (1)

(11.20)

The following various equations of the function g ( χ ) have been proposed.

256

11 Constitutive Equations of Soils

1 ) Original Cam-clay model (Schofield and Wroth, 1968)

g = exp( χ )

(11.21)

pcr = F / e

(11.22)

2 ) Modified Cam-clay model (Burland, 1965; Roscoe and Burland, 1968) g = 1+ χ 2

(11.23)

pcr = F / 2

(11.24)

3 ) Hashiguchi model (Hashiguchi, 1972) g = exp (

χ2 2

(11.25)

)

pcr = F/ e

(11.26)

The above-mentioned three yield surfaces are shown in Fig. 11.4. Among them, only the yield surface of the modified Cam-clay model in Eq. (11.23) does not involve the corner so that the singularity of the normal of the surface is not induced. Equation (6.30) with Eqs. (11.11) and (11.23) is rewritten as

{

p − ( F / 2) 2 ||σ' || 2 + =1 } ( M F /2) 2 F /2

(11.27)

Compression

q

Hashiguchi (1972) model Modified Cam-clay model Original Cam-clay model

Mc

0

1

F Me e

F 2

F

F

p

e Extension

Fig. 11.4 Various yield surfaces of soils

11.2 Yield Conditions

257

Here, consider further the yield surface fulfilling the following conditions. 1) It involves not only positive but also negative pressure ranges. Here, note that the subloading surface is indeterminate at the null stress point in the initial subloading surface model if the normal-yield surface passes thorough at the null stress point in which the singular point of plastic modulus is induced, while this problem is not induced in the extended subloading surface model because the similarity-center of the normal-yield and the subloading surfaces is not fixed at the null stress point. The exclusion of the singularity of plastic modulus at the null stress point is of important for the engineering design of soil structures because soils near the side edges of footings, soils at the pointed ends of piles, etc. are exposed to the null or further negative stress state. In addition, the inclusion of tensile yield strength is of importance for the engineering design of structures of natural soils such as soft rocks and cement-treated soils widely used recently, which have the tensile yield strength. 2) In the case that the anisotropy does not change, the yield surface expands/contracts keeping the similarity with respect to the origin of stress space so that the yield stress increases/decreases in all directions in this space. 3) For the sake of mathematical simplicity, the yield condition is described by a separate form consisting of the function of the stress and internal variables f (σ, β) and the function involving the isotropic hardening variable, i.e. the isotropic hardening function F ( H ) , which describes the size of the yield surface. Here, the function f (σ, β) must be a homogeneous function of the stress tensor σ in order to fulfill the above-mentioned conditions 2 ) and 3 ). Equation (11.27) becomes the following equation through the translation to the negative pressure range by ξ F ( p → p + ξ F ) (Hashiguchi and Mase, 2007).

{

p − {(1 / 2) − ξ }F 2 || σ' || 2 + =1 } F /2 ( MF / 2) 2

(11.28)

where ξ is the material constant, while it must fulfill 0 ≤ ξ ≤ 1/2 since the tensile yield stress is smaller than the compression stress and ξ < pe / F since the volume does not become infinite by the elastic deformation inside the yield surface, i.e. for p > −ξ F . The yield surface in Eq. (11.28) is depicted in Fig. 11.5 for the axisymmetric stress state, where M is given from the relation || σ' || = MF/ 2 = M (1 − 2ξ ) F/ 2 as follows:

M=

1 M 1 − 2ξ

(11.29)

258

11 Constitutive Equations of Soils

φc is described from Eqs. (11.13) and (11.29) as follows: φc = sin −1 (

3M c 2 6 + Mc

) = sin − ( 2 1

3(1 − 2ξ ) Mc 6 + (1 − 2ξ ) M c

)

(11.30)

where Mc and M c are the values of M and M in the axisymmmetric compression stress state. q is the axial difference stress, i.e. q ≡ σ l − σ a ( σ a : axial stress

，

σ l : lateral stress) in Fig. 11.5. v

Normal-consolidation line Critical state line Swelling line

p

0 q

ξˆ MF 2

1 M

M ( =

1 − pe

0

1 M) 1 − 2ξ

(1/ 2 − ξ )F

p

F /2

F

− ξ F (> − pe )

py (= (1 − ξ )F )

Fig. 11.5 Yield surface of soils with tensile strength

Equation (11.28) can be transformed to the equation

(1 − ξ )ξ F 2 + (1 − 2ξ ) pF − ( p 2 + χ 2 ) = 0

(11.31)

11.3 Isotropic Hardening Function

where

259

χ ≡ || σ' ||

(11.32)

M

Equation (11.31) can be expressed in the separated form of the function f ( p, χ ) of the stress and internal variables and the hardening function F , i.e. ⎧ p{1 + ( χ / p ) 2 } for ξ =0

f ( p, χ ) = F , f ( p , χ ) ⎪ = ⎨1

p ⎪ ξ ( χ − ξ p ) for ξ ≠ 0 ⎩

(11.33)

where ξ ≡ 2(1 − ξ )ξ , ξ ≡ (1 − 2ξ ) , pχ ≡ p 2 + 2ξ χ 2

(11.34)

In the above, the yield surface of soils is formulated so as to fulfill the conditions 1 )-3 ) based on the modified Cam-clay model. It is difficult to derive the other yield surface fulfilling the conditions 1 )-3 ). For instance, consider the translation of the original Cam clay model to the negative pressure range by p → p + ξ F . ( p + ξ F ) exp (

|| σ' || p +ξF / M) = F

(11.35)

However, a separated form of the stress and internal variables and the hardening function cannot be derived from this equation. On the other hand, the translation of the yield surface to the negative pressure range by the constant value C y ( p → p + C y) is adopted for constitutive equations for unsaturated soils (e.g. Alonso et al., 1990; Simo and Meschke, 1993; Borja, 2004). The modified Cam-clay model, for instance, is described by this translation as follows:

{

p − (1 / 2) F + C y 2 ||σ' || 2 + } =1 F /2 ( MF / 2) 2

(11.36)

In this equation, the yield surface expands/contracts from/to the fixed point σ = C y I ( p = C y ) on the hydrostatic axis and thus it does not fulfill the condition 2 ). The incorporation of this yield condition into the subloading surface model leads to the physical impertinence that the unloading is induced against the fact a large plastic deformation is induced when the stress translates towards the negative pressure direction.

11.3 Isotropic Hardening Function The isotropic hardening/softening of soils is induced substantially by the plastic volumetric strain. Substituting py = (1 − ξ ) F into Eq. (11.3)2 in accordance with

260

11 Constitutive Equations of Soils

the translation of the yield surface by ξ F to the negative pressure range based on the discussion in 11.2, one has −ξ +p ε vp = −( ρ − γ ) ln (1 ) F e ξ − (1 ) F0 + pe

(11.37)

Here, selecting H as •

H ≡ −ε vp , H ≡ − Dvp , h ( σ, H i , N) = −t rN

(11.38)

so that F increases with H , the hardening function F ( H ) is given from Eq. (11.37) as follows: pe p H F ( H ) = (F0 + −e ) exp ρ γ − 1 − ξ 1 ξ −

(11.39)

pe dF 1 F ' ≡ dH = ρ − γ ( F + 1 − ξ )

(11.40)

While the plastic volumetric strain is dominant for the hardening/softening of soils, the deviatoric strain would induce the remarkable softening in the negative range of pressure. Then, extend the evolution rule of the hardening/softening function H in Eq. (11.38) as follows: •

p H = − Dvp − ζ || D ' || 〈− p〉, h ( σ, H i , N) = −t rN − ζ || N' || 〈− p〉

(11.41)

where ζ is the material constant. On the other hand, the following isotropic hardening rule has been used for sands, while the isotropic hardening and softening are induced by the deviatoric strain rate when the stress ratio is higher and lower, respectively, than a certain value M d (Nova, 1977; Wilde, 1977; Hashiguchi and Chen, 1998). • p || || H = − Dvp + μ d || D p' || σ' − Md F

Md ( cos3θ σ ) ≡

14 6 sin φd (3 − sin φd )(8 + cos3θ σ )

(11.42) (11.43)

where μ d and φd are the material constants. The stress paths for sands under the constant volume or undrained condition can be predicted realistically by the isotropic hardening rate in Eq. (11.42) with Eq.(11.43), in which the remarkable stress rise and drop are induced in loose and dense sands, respectively (see Fig. 11.6). On the other hand, these behavior can be predicted also by introducing the super-yield surface as will be described in 11.10.

11.4 Rotational Hardening

q t ica Cri

261

ate l st

line

Stress path

q

a it ic Cr

Stress path

line

Initia l yield surface

Initial yield surface

0

ate l st

p Loose sands

0

p Dense sands

Fig. 11.6 Stress paths under constant volume or undrained condition

11.4 Rotational Hardening The inherent anisotropy represented in the orthotropic anisotropy described in 10.1.4 cannot be ignored in metals and woods. On the other hand, the induced anisotropy is more dominant in soils since soils are assemblies of particles with weak cohesions between them and thus the rearrangement of soil particles is induced easily. Here, the yield surface of soils must always involve the origin of stress space but does very slightly because of the high frictional property. Note that the stress can never return to the origin undergoing the remarkable softening (contraction of the yield surface) following the plastic potential flow rule, once the yield surface translates so as not to involve the origin, as illustrated on the (p, q) plane for the axisymmetric stress state in Fig.11.7. Therefore, the kinematic hardening (Prager, 1956) is not applicable to soils. On the other hand, the anisotropy of soils with the frictional property, i.e. the pressure-dependence of yield surface can be described by the concept of the rotational hardening that the yield surface rotates around the origin of stress space by anisotropy. Sekiguchi and Ohta (1977) proposed the replacement of the deviatoric

q

∂f ∂ı

0

tr

dı ı

∂f ∂f ( = − p ) > 0 : softening ı ∂ ∂

Yield surface: contraction p

Fig. 11.7 Inadequacy of kinematic hardening for description of anisotropy of soils

262

11 Constitutive Equations of Soils

stress σ' to the novel variable σ' − pβ , where β is called the rotational hardening variable. Then, the yield condition in Eq. (11.28) or (11.33) is extended as follows (Hashiguchi and Mase, 2007):

{

p − {(1 / 2) − ξ }F 2 || ||2 + σ' 2 = 1 } F /2 ( M F / 2)

(11.44)

or

⎧ p{1 + ( χ / p) 2 } for ξ =0 χ f ( p, ) = F , f ( p, χ ) = ⎪ ⎨1 p ⎪ ξ ( χ − ξ p) for ξ ≠ 0 ⎩ where

(11.45)

σ ' ≡ σ' − pβ

(11.46)

χ ≡ || σ' ||

(11.47)

M

M (cos3θ σ ) =

14 6 sin φc (3 − sin φc )(8 + cos3θ σ )

cos 3θ σ ≡ 6t r τ 3 ,

(11.48)

τ ≡ ||σσ' ||

(11.49)

'

The yield surface in Eq. (11.44) or (11.45) is depicted in Fig. 11.8.

q Compression

tra l C en

a xis

tan −1 ( 32 β a )

−ξF

0

py (= (1 − ξ )F )

Extension

Fig. 11.8 Rotated yield surface in the (p, q) plane

p

11.4 Rotational Hardening

263

The evolution rule of rotational hardening tensor β is described below (Hashiguchi and Chen, 1998; Hashiguchi, 2001). The following assumptions are adopted for the formulation of the evolution rule. 1 ) Rotation of the yield surface is induced only by the deviatoric component of the plastic strain rate independent of the mean component. 2 ) The rotation ceases when the central axis of yield surface reaches the surface, called the rotational hardening limit surface, which exhibits the conical surface having the summit at the origin of stress space. Let the rotational hardening limit surface be given by

η = Mr

(11.50)

where M r is the stress ratio in the rotational hardening limit surface, called the rotational hardening limit stress ratio, and let it be given by M r (cos3θ σ ) =

14 6 sin φ r (3 − sin φ r )(8 + cos3θ σ )

(11.51)

φ r being the material constant, called the rotational hardening limit angle. 3 ) The central axis of yield surface η = β rotates towards the conjugate line β = M r τ on the rotational hardening limit surface, where the conjugate line is the generating line of the rotational hardening limit surface which is observed from the hydrostatic axis in the same direction observed from the central axis η = β of the yield surface to the current stress (see Fig. 11.9). 4 ) In the proportional loading process, the central axis of the normal-yield surface approaches the stress path in the proportional loading process. Further, the magnitude of rotational rate β is larger for a larger distance from the central ||. This assumption is axis to the current stress and thus it is the function of || ı ' necessary also to exclude the singularity of the rotational direction in the state that the stress lies on the central axis of the normal-yield surface (see Fig. 11.10). Based on the above-mentioned assumptions, let the following evolution rule of rotational hardening be postulated.

|| σ ' || || || ( Mrτ − β) β = br || D p' || σ ' ( M r τ − β ) , b (σ, H i ) = br F F (11.52) where b r is the material constant and M r is given by the following equation in an identical form to that of Eq. (11.13).

264

11 Constitutive Equations of Soils

Yield surface

Central axis Ș= ȕ

σ3

Ĳ ı

D ȕ Ĳ

Hydrostatic axis Ș=0

Rotational hardening limit surface || Ș|| = M r Conjugate generating line Ș = MrĲ σ2

0

σ1

Fig. 11.9 Direction of rotation of yield surface (illustrated in the principal stress space)

Compression Yield surface

q

Ro

tat

al io n

r ha

n de

in g

a ur f ti s li m

ıa ıb

ce a x is t ra l n e D a ce s ur f ȕa = 0 C d l e i of y D ȕb = 0 Singular point of rotational direction

p

0

R ota

t io na

l ha r de n i n

g li m it sur fa ce

Extension

Fig. 11.10 Singularity in direction of rotation of yield surface on central axis of yield surface

11.5 Extended Subloading Surface Model

M r (cos3θ σ ) =

265

14 6 sin φ r (3 − sin φ r )(8 + cos3θ σ )

(11.53)

The relation between the axial components of the rotational hardening variable vs. the deviatoric plastic strain in the axisymmetric stress state under F = const. for sake of simplicity is depicted in Fig. 11.11. It is identical to the kinematic hardening variable shown in Fig. 10.1. βa 2M 3 r 3 b | η | ( 2 M −β ) 3 r a 2 r a

1

3 b | η | ( 2 M + β ) a 3 r 2 r a

0

1

ε ap'

− 2 Mr 3

Fig. 11.11 Relation of axial component of rotational hardening variable vs. axial plastic strain in the axisymmetric stress state

11.5 Extended Subloading Surface Model The extended subloading surface for the normal-yield surface in Eq. (11.45) is described by replacing σ into σ ≡ σ − α as follows:

⎧ χ 2 ⎪ p{1 + ( p ) } for ξ =0 f ( p, χ ) = RF , f ( p, χ ) = ⎪ ⎨ ⎪ 1 ( p − ξ p ) for ξ ≠ 0 ⎪⎩ ξ χ

(11.54)

p 2 + 2ξ χ 2

(11.55)

where pχ ≡

σ' ≡ σ' − pβ

(11.56)

266

11 Constitutive Equations of Soils

|| || ， M ( cos 3θ σ ) = χ ≡ σ' M

14 6 sin φ c (3 − sin φ c )(8 + cos3θ σ )

cos3θ σ ≡ 6t r τ 3 ， τ ≡ σ ' ||σ ' ||

σ ij'

(11.57)

)

(11.58)

α ≡ s − Rs = (1 − R )s, σ (= σ − (1 − R )s ) = σ + R s

(11.59)

(τ ij ≡

σ rs' σ sr'

Because of α = 0 in soils, it holds from Eq. (8.3) that

The normal-yield and the extended subloading surfaces for the yield surface for Eq. (11.44) or (11.45) are depicted in Fig. 11.12.

Normal- yield surface

q

Subloading surface i gl n in e d ı har al n o i ta t Ro

ts mi

a ur f

ce

Compression

− (β a − β l )(1− ξ ) F = − 3 (1 − ξ ) β a F 2

(

)

s − tan −1 (32 β a)

−ξ F

(1 − ξ ) F

0

Rota

t io na

l har

d en i ng li mit

p

Extension

s ur fa

ce

Fig. 11.12 Rotated normal-yield and extended subloading surfaces in the (p, q) plane

The rotational hardening rule is given from Eqs. (11.52) with the replacement of

σ to σ ≡ σ − α by β = br || D p' ||

|| σ ' || || σ || ( M r τ − β ) , b (σ, H i ) = br ' ( M r τ − β ) F F

(11.60)

Further, setting α = 0 , the plastic modulus is obtained from Eq. (8.27) as follows: ( ) M p ≡ tr N F ′h( σ, H i ; N) σˆ +{ U R + c( 1 − 1) }σ F R R

[{

|| || ∂f (σ, β ) b (σ, H i )|| N' ||) σ − N' t r( RF ∂β

(

11.6 Partial Derivatives of Subloading Surface Function

− R (1 − R )t r (

∂f (sˆ, β) b (σ, H i )) sˆ ∂β

267

)}] = 0

(11.61)

where the functions h and b are given by Eqs. (11.41) and (11.60).

11.6 Partial Derivatives of Subloading Surface Function The partial derivatives of the function in Eq. (11.54) are shown below.

⎧1 + ( χ / p ) 2 + p (−2 p −3 )χ 2 for ξ =0 p ∂f ( , χ ) ⎪ = ⎨1 1 2p ∂p ⎪ ξ 2 p − ξ for ξ ≠ 0 χ ⎩

(

)

⎧ χ 2 ⎪1 − ( p ) for ξ =0 ⎪ =⎨ ⎪ 1 p − ξ for ξ ≠ 0 ⎪⎩ ξ pχ

(

)

⎧ χ ⎧ χ for ξ =0 ⎪2 p for ξ =0 ⎪2 ∂f ( p, χ ) ⎪ ⎪ p =⎨ =⎨ ∂χ ⎪ 1 1 4ξχ for ξ ≠ 0 ⎪2 χ for ξ ≠ 0 ⎪ ξ 2 pχ ⎪ pχ ⎩ ⎩ ∂p = −1I 3 ∂σ ∂ σ' = I − 1 I ⊗ I = I' 3 ∂σ

( ∂∂σσ ' ij

kl

=

∂ (σ ij + pδ ij ) 1 = (δ ik δ jl + δ il δ jk ) − 1 δ ijδ kl 3 2 ∂σ kl

)

∂ σ' = I − 1 I ⊗ I + 1 β ⊗ I = I' + 1 β ⊗ I 3 3 3 ∂σ

( ∂∂σσ '

ij

kl

=

∂ (σ 'ij − p β ij ) 1 = (δ ik δ jl + δ il δ jk ) − 1 δ ijδ kl + 1 βijδ kl 3 2 3 ∂σ kl 14 6 sin φ c ∂M = − ∂ cos3θ σ (3 − sin φ c )(8 + cos3θ σ ) 2

)

268

11 Constitutive Equations of Soils

=−

M (= − 3 − sin φ c M 2 ) 8 + cos3θ σ 14 6 sin φ c

∂τ = 1 ( I − τ2) ∂ σ' || σ'||

σ ij'

∂

( στ ' ∂ ∂

=

ij

=

σ rs' σ sr' =

∂σ 'kl

kl

∂σ 'ij ∂ σ rs' σ sr' σ rs' σ sr' − σ ij' ∂σ 'kl ∂σ 'kl

σ rs' σ sr'

1 (δ δ + δ δ ) σ σ −σˆ ˆ 'ijτ kl ' sr' rs il jk 2 ik jl

=

σ rs' σ sr' 1

σ rs' σ sr'

)

{12 (δikδ jl + δilδ jk ) −τˆijτˆkl} ∂(t r τ 3) = 3 2 τ ∂τ

(∂τ ∂ττ τ

rs st tr ij

= δ irδ jsτ stτ tr + τrs δ isδ jtτ tr + τrsτ stδ itδ jr = 3τ irτ rj

)

∂cos3θ σ = 3 ( 6 τ 2 − cos3θ σ τ ) || σ' || ∂ σ'

(∂

∂τ 6τ lmτ mnτ nl ∂τ τ τ ∂τ = 6 lm mn nl rs = 3 6τ s nτ nr rs ∂ τ rs ∂σ 'ij ∂σ 'ij ∂σ 'ij = 3 6τ s nτ nr

σ pq ' σ qp '

1

=3 6

=

1

σ pq ' σ qp ' 3

σ pq ' σ qp '

{12 ( δriδ sj + δ rjδsi ) − τ rsτ ij }

(τ inτ n j −τ s nτn rτ rsτ ij )

)

( 6τ inτ nj − cos3θ σ τ ij)

11.6 Partial Derivatives of Subloading Surface Function

269

∂χ 3 = 1 τ+ ( 6 τ 2 − cos3θ σ τ )} 8 + cos3θ σ ∂ σ' M

{

( ∂∂σχij' =

∂χ

∂ σ pq ' σ qp '

∂ σ pq ' σ qp '

∂σ ij'

+

∂χ

∂ M ∂ cos3θ σ ∂M ∂cos3θ σ ∂σ ij'

∂ σ pq ' σ qp ' σ ij' ∂M ∂cos3θ σ = 1 − 2 ∂cos3θ σ ∂σ ij' M ∂ σ pq M ' σ qp '

{

3 = 1 τ ij + ( 6τ irτ rj − cos3θ σ τ ij )} 8 cos + 3θ σ M

)

∂f ( p, χ ) ∂f ( p, χ ) ∂ p ∂f ( p, χ ) ∂χ ∂ σ' = −1 + 3 ∂ p ∂σ ∂σ ∂χˆ ∂ σ' ∂σ ∂f ( p, χ ) ∂f ( p, χ ) ∂χ = −1 ( I' + 13 β ⊗ I) I+ 3 ∂p ∂χ ∂ σ' = −1 3

∂f ( p, χ ) ∂f ( p, χ ) I+ ∂p ∂χ

[ ∂∂σχ' − 13tr{ ∂∂σχ' ( I − β )}I]

( ∂f ∂( σp,ijχ ) = − 13 ∂f (∂pp, χ ) ∂∂σpij + ∂f (∂pχ, χ ) ∂∂σχ

∂σ rs ' σ ∂ ij ' rs

= −1 3

∂f ( p, χ ) ∂ p ∂f ( p , χ ) ∂χ + ∂ p ∂ σij ∂σ rs ∂χ '

{12 (δ riδ sj + δ rjδ si ) − 13 δ rsδij + 13 βrsδij} = −1 3

∂f ( p , χ ) ∂f ( p , χ ) ∂χ δ ij + ∂p ∂χ ' ∂σ rs

{12 (δ riδ sj + δ rjδ si ) − 13 δ rsδij + 13 βrsδij})

(11.62)

270

11 Constitutive Equations of Soils

⎧ 1 χ 2 χ ∂χ 1 ∂χ − tr{ ( I − β )}I ⎪− 3{1 − ( p ) } I + 2 p 3 ∂ σ' ∂ σ' ⎪ for ξ =0 ∂f ( p, χ ) ⎪⎪ =⎨ ∂σ ⎪− 1 1 p −ξ I + 2 χ ∂χ − 1 tr{ ∂χ ( I − β )}I ⎪ 3 ξ pχ p χ ∂ σ' 3 ∂ σ' ⎪ for ξ ≠ 0 ⎪⎩ (11.63)

]

[

(

]

[

)

∂f ( p, χ ) ∂f ( p , χ ) ∂χ = −p ∂β ∂χ ∂σ'

( ∂f ∂( pβ, χ ) = ∂f (∂pχ, χ ) ij

= −p

∂χ ∂σ rs ' = ∂f ( p, χ ) ∂χ (− p I ) rsij ∂χ ∂σ rs ∂σ rs ' ∂ βij '

∂f ( p, χ ) ∂χ 1 ∂f ( p , χ ) ∂χ (δ rjδ sj + δ rjδ si ) = − p 2 ∂χ ∂χ ∂σ rs ' ∂σ rs '

)

⎧ ∂χ for ξ =0 ⎪−2 χ ∂ σ' ∂f ( p, χ ) ⎪ =⎨ ∂β ⎪−2 p χ ∂χ for ξ ≠ 0 ⎪ pχ ∂σ' ⎩

(11.64)

⎧ ps {1 + ( χs / ps ) 2 } for ξ =0 ⎪ f (sˆ, β) = f ( ps , χs ) = ⎨ 1 p ⎪ ξ ( χs − ξ ps ) for ξ ≠ 0 ⎩

(11.65)

where

ps ≡ − 1 tr s, s' ≡ s + ps I 3

(11.66)

s ' ≡ s' − ps β

(11.67)

χs ≡ || s' ||

(11.68)

14 6 sin φc (3 − sin φc )(8 + cos 3θ s)

(11.69)

Ms

M s (cos3θ s ) =

11.7 Calculation of Normal-Yield Ratio

271

cos 3θ s ≡ 6t r τˆ s3 , τ s ≡ s' || s' ||

∂ χs 3 = 1 τ + ( 6 τs2 − cos3θ s τs )} ∂ s' M s s 8 + cos3θ s

{

⎧ χ ∂ χs ⎪−2 s ∂ s' for ξ =0 ∂f (sˆ, β) ⎪ =⎨ χ ∂β p ⎪−2 s χs ∂ s for ξ ≠ 0 ∂ s' ⎪ pχs ⎩

(11.70) (11.71)

(11.72)

11.7 Calculation of Normal-Yield Ratio The normal-yield ratio R can be calculated directly by R = f (σ,β)/F in the initial subloading surface model but it has to be obtained by the numerical calculation in the extended subloading surface model as will be described below. First, one has

p = − 1 tr(σ + Rs) = −(σ m + Rsm ) , σ' = σ' + Rs' 3

(11.73)

from Eq. (8.6). Substituting Eq. (11.73) into Eq. (11.56) and (11.57), it is obtained that 1 (11.74) σ' ( = σ' − p β ) = σ' + R s' + 3{tr(σ + R s)}β = σ' + R s' + (σ m + R sm ) β 1 || || || σ' + R s' + 3{tr(σ + R s)}β || (11.75) χ ≡ σ' = M M Further, substituting Eq. (11.73)-(11.75) into Eq. (11.54) of the extended subloading surface, one has the following equation and can transform it in turn. ⎫ ⎪ 2 ⎪ = RF for ξ =0 M −(σ m + Rsm ) 1 + ⎪ −(σ m + R sm ) ⎪ ⎪ ⎬ ⎪ || + (σ m + Rsm ) β || 2 ' s 1 {−(σ + Rs ) 2 ' + R σ + ξ (σ m + R sm ) ⎪ m m } + 2ξ { } ξ ⎪ M ⎪ = RF for ξ ≠ 0 ⎪ ⎪⎭

{ (

[

||σ' + Rs' + (σ m + Rsm ) β ||

)}

]

(11.76)

272

11 Constitutive Equations of Soils

||(σ ∗ + R s∗) + (σ m + Rs m )β || − (σ m + Rs m ) 2 − Mˆ

⎫ + Rs m )RF ⎪ ⎪ for ξ =0 ⎪⎪ ⎬ || 2 + ( σ 2 s + || β ) R s ' ' + ⎪ m R σ m {−(σ m + Rs m )} + 2ξ{ } ⎪ M ⎪ ( σ ⎪⎭ ≠ for ξ 0 + s R ) ξ m − ξ m = RF

{

2

} = (σ

m

2 2 M 2 (σ m + Rs m ) + t r(σ' + R s' ) + 2(σ m + Rs m )t r{(σ' + R s' )β}

⎫ ⎪ + (σ m + Rs m ) 2 || β ||2 +M 2 (σ m + Rs m )RF = 0 for ξ = 0 ⎪⎪ ⎬ M 2 (σ m + Rs m ) 2 + 2ξ t r( σ' + Rs' ) 2 + 4ξ (σ m + Rs m )t r{(σ' + R s' )β} ⎪ ⎪ + 2ξ (σ m + Rs m )2 || β || 2 − [ M {ξ RF − ξ (σ m + R sm )}] 2 = 0 for ξ ≠ 0 ⎭⎪ ⎫ 2 2 2 2 M 2σ m + 2 M σ m s m R + M sm R 2 + || σ' || 2 + 2t r(σ' s') R +|| s' || 2 R 2 ⎪ ⎪ + 2(σ m + Rs m )t r(σ' β) + 2(σ m R + s mR 2 )t r( s' β) ⎪ ⎪ 2 2 2 2 2 2 + (σ m + 2 R σ m s m + sm R )|| β || +M ( σ m FR + s m FR ) = 0 ⎪ ⎪ for ξ = 0 ⎪ ⎪ 2 2 2 2 2σ 2 + 2 + || || 2 ξ σ' ⎬ M m 2 M σ m s m R + M sm R ⎪ 2 + 4ξt r(σ' s') R + 2 ξ|| s' || R 2 + 4ξ(σ m + s mR)t r(σ' β) ⎪ ⎪ + 4ξ(σ m R + s mR 2 )t r( s' β) + 2ξ (σ m 2 + 2 σ m s mR + sm2 R 2 )|| β || 2 ⎪ ⎪ ⎪ − M 2{( ξ F −ξ s m ) 2 R 2 − 2 ξ σ m (ξ F −ξ s m ) R + ξ 2σ m2 } = 0 ⎪ for ξ ≠ 0 ⎪ ⎭

⎛ ⎜ ⎝

{ξRF − ξ (σ m + Rsm )}2 = {(ξF −ξ sm) R − ξσ m }2 = (ξF −ξ sm) 2 R 2 − 2 ξσ m (ξF −ξ sm ) R + ξ σ m2 2

⎞ ⎟ ⎠

11.7 Calculation of Normal-Yield Ratio

273

2 2⎫ M 2 sm2 R 2 + || s' || 2 R 2 + 2s mt r( s' β) R 2 + sm2 || β ||2 R 2 + M s m FR ⎪ ⎪ + 2 M 2σ m s m R + 2t r(σ' s') R + 2 s mt r(σ' β) R ⎪ ⎪ 2 2 + + 2σ m t r( s' β) R M σ m FR + 2 σ ms m || β || R ⎪ ⎪ 2 2+ 2 2 + M σ m || σ' || +2σ m t r(σ' β) + σ m2 || β || = 0 for ξ = 0 ⎪ ⎪⎪ 2 2 2 2 2 2 2 2 M sm R 2 + 2 ξ || s' || R + 4ξ s mt r( s' β) R + 2ξ sm || β || R ⎬ ⎪ 2 2 2 2 − M R (ξ F −ξ s m ) + 2M σ m s m R + 4ξ t r(σ' s')R ⎪ ⎪ 2 + 4ξ s mt r(σ' β) R + 4ξσ m t r( s' β) R + 4 ξ σ ms m|| β || R ⎪ ⎪ + 2M 2ξ σ m (ξ F −ξ s m ) R +M 2σ m2 + 2ξ || σ' ||2 + 4ξσ m t r(σ' β) ⎪ ⎪ 2 2 ⎪ + 2ξσ m2 || β || − M 2ξ 2σ m = 0 for ξ ≠ 0 ⎪⎭

⎫ ⎪ ⎪ 2 + {2M σ m s m + 2 t r(σ' s') + 2s mt r(σ' β) + 2σ m t r( s' β) ⎪ ⎪ 2 2 2 2 2 + 2 σ ms m || β || +M σ m F }R + M σ m + || σ' || ⎪ ⎪ + 2σ m t r(σ' β) + σ m2 || β ||2 = 0 for ξ = 0 ⎪ ⎬ 2 2 2 2 2 {M 2 sm2 + 2 ξ || s' || +4ξ s mt r( s' β) + 2ξ sm || β || − M 2(ξ F −ξ s m ) }R ⎪ ⎪ 2 ⎪ + ) + + 2{M σ m s m + 2 ξ t r(σ' s' 2ξ s mt r(σ' β) 2ξσ m t r( s' β) ⎪ ⎪ +2 ξ σ ms m|| β ||2 + M 2ξ σ m (ξ F −ξ s m )}R + M 2 (1 − ξ 2)σ m2 ⎪ 2 2 ⎪ + 2 ξ || σ' || + 4ξσ m t r(σ' β) + 2ξ σ m2 || β || = 0 for ξ ≠ 0 ⎭

2 {M 2 sm2 + || s' || 2 + 2 s mt r( s' β) + sm2 || β || 2 + M s m F }R 2

Solving this quadratic equation, the normal-yield ratio R is expressed as follows: ⎧ B 2 − AC − B for ξ = 0 ⎪ A ⎪ R=⎨ 2 ⎪ B − AC − B for ξ ≠ 0 ⎪⎩ A

(11.77)

274

11 Constitutive Equations of Soils

where

⎫ ⎪ ⎪ 2 B ≡ 2 M σ m s m + 2t r(σ' s') + 2 s mt r(σ' β) + 2σ m t r( s' β) ⎪ 2 ⎪ 2 + 2 σ m s m || β || +M σ m F ⎪ ⎪ 2 2 2 2 2 C ≡ M σ m + || σ' || +2σ m t r(σ' β) + σ m || β || ⎪ ⎪ 2 β 2 2 2 2 A ≡ M sm + 2 ξ || s' || +4ξ s mt r( s' β) + 2ξ sm || || ⎬ ⎪ − M 2(ξ F −ξ s m ) 2 ⎪ ⎪ B ≡ M 2σ m s m + 2 ξ t r(σ' s') + 2ξ s mt r(σ' β) + 2 ξσ m t r( s' β) ⎪ ⎪ 2 2 + 2 ξ σ ms m|| β || + M ξ σ m (ξ F −ξ s m ) ⎪ ⎪ 2 2 2 2 C ≡ M 2 (1 − ξ 2)σ m + 2 ξ || σ' || + 4ξσ m t r(σ' β) + 2ξ σ m || β || ⎪ ⎭⎪ 2 A ≡ M 2 sm2 + || s' || 2 + 2s mt r( s' β) + sm2 || β ||2 + M s m F

(11.78)

Explicit calculation processes: 1 ) First step (beginning of calculation): Calculate the normal-yield ratio R by Eq. (11.77), substituting the trial value M = 2 6sinφ c /3

which is the average of M = 2 6sinφ c /(3 + sinφ c ) and

2 6sinφ c /(3 − sinφ c ) .

2 ) After second step: Recalculate R by substituting the value

M ( cos3θ σ ) (=

14 6 sin φ c 14 6 sin φ c )= (3 − sin φ c )(8 + cos3θ σ ) (3 − sin φ c )(8 + 6t r τ 3) (11.79)

into Eq. (11.77), while the value of R obtained in the former step is used in Eq. (11.79).

11.8 Simulations of Test Results

275

3 ) Repeat the process 2 ) until R will reaches the convergence. • Note here that R is calculated by the accumulation of R + R dt with • p R = U ( R ) || D || in the loading process but one has to calculate R by the above-mentioned processes in the unloading process. d

11.8 Simulations of Test Results Some simulations of test data are given below in order to show the capability of the subloading surface model to reproduce the real deformation behavior of soils (Hashiguchi and Chen, 1989). Hereafter, all stresses exhibit effective stresses, i.e. stresses excluded pore-pressure. The simulation of the test data (after Saada and Bianchini, 1989) for Hostun sand subjected to the drained triaxial compression with a constant lateral stress, which includes the unloading-reloading process, is shown in Fig. 11.13 where the material constants and the initial values are selected as follows: Material constants: Yield surface (ellipsoid): φc = 27 , ⎧ ⎧⎪ volumetric : ρ = 0.008, γ = 0.003, pe = 10 kPa, ⎪isotropic ⎨ Hardening/softening ⎨ ⎩⎪deviatoric : μ d = 0.6, φd = 25 , ⎪ ⎩ rotational : br = 10, φr = 20 , Evolution of normal - yield ratio : u1 = 1.5, m1 = 3.8, Translationon of similarity - center : c = 20, Elastic shear modulus : G = 200 000 kPa,

Initial values:

Hardening function: F0 = 400 kPa, Rotational hardening variable : β 0 = 0, Center of similarity : s 0 = −50I kPa, Stress : σ 0 = − 100I kPa where the equation U ( R ) = u1 (1/ R

m1

− 1) is used for the evolution rule of the nor-

mal-yield ratio in Eq. (7.13). The simulation of the test data (after Saada and Bianchini, 1989) for Hostun sand subjected to the drained proportional loading with b (=(σ 2 − σ 3 )/(σ 1 − σ 3 )) = 0.666 (θσ = 19 09' ) from σ 0 = −500I kPa by the true triaxial test apparatus is shown in

276

11 Constitutive Equations of Soils

Fig. 11.14. The material parameters are the same as those for the above-mentioned drained triaxial compression, while the sample was preliminarily loaded the isotropic compression from σ = −100I kPa to −500I kPa before the test.

||ı' || p

|| ³ D' dt ||

εv

|| ³ D' dt || Fig. 11.13 Drained behavior of Hostun sand (data from Saada and Bianchini, 1989). Measured and calculated results are shown by the dashed and solid lines, respectively.

11.8 Simulations of Test Results

277

|| ı' || p

||³ D' dt ||

εv

||³ D' dt || Fig. 11.14 Drained proportional loading behavior of Hostun sand (data from Saada and Bianchini, 1989). Measured and calculated results are shown by the dashed and solid lines, respectively.

278

11 Constitutive Equations of Soils

|| ı' || p

||³ D ' dt || Fig. 11.15 Undrained behavior of Banding sand (data from Castro, 1969). Calculated results are shown by the solid lines

11.8 Simulations of Test Results

279

The simulations of the test data (after Castro, 1969) for Banding sand subjected to the undrained triaxial compression with a constant lateral stress are shown in Fig. 11.15 where the material constants and the initial values are selected as follows: Material constants: Yield surface (ellipsoid): φc = 26, 30, 31, 32 , ⎧ ⎧ ⎧ ρ = 0.025, 0.018, 0.014, 0.010, ⎪ ⎪ ⎪ ⎪ ⎪ volumetric ⎨γ = 0.0067, 0.0065, 0.0060, 0.0058, ⎪ pe = 0, 10, 30, 80 kPa, ⎪isotropic ⎪ ⎩ ⎨ ⎪ Hardening/softening ⎨ ⎪ ⎧ μ = 1.00, 0.65, 0.30, 0.10, ⎪ ⎪deviatoric ⎪⎨ d ⎪ ⎪⎩ ⎪⎩φd = 40, 33, 30, 20 , ⎪ ⎪⎩rotational : br = 10, φr = 20 , ⎧u1 = 0.1, 0.3, 0.5, 1.0, 33.0, Evolution of normal - yield ratio ⎨ ⎩m1 = 0.1, 0.4, 0.5, 0, 7, Translationon of similarity - center : c = 20, 18, 14, 8, Elastic shear modulus : G = 18 000, 23 000, 25 000, 35 000 kPa,

Initial values:

Hardening function: F0 = 410, 480, 520, 580 kPa, Rotational hardening variable : β 0 = 0, Center of similarity : s0 = −200, −110, −100, − 80 I kPa, Stress : σ 0 = − 67.0I kPa where

the

four

values correspond to the initial relative densities Dr = 0.27, 0.44, 0.47, 0.64 , respectively, in this order. The simulation of the test data (after Ishihara et al., 1975) for the cyclic mobility of loose Niigata sand (e0 = 0.737) with the constant stress amplitude q = ±0.71 kgf/cm2 under the undrained condition is depicted in Fig. 11.16 where the material constants and the initial values are selected as follows: Material constants: Yield surface (ellipsoid): φc = 28 , ⎧ ⎧⎪ volumetric : ρ = 0.01, γ = 0.0065, pe = 0.05 kPa, ⎪isotropic ⎨ Hardening/softening ⎨ ⎩⎪deviatoric: μ d = 1.0, φd = 35 , ⎪ ⎩rotational : br = 10, φr = 20 , Evolution of R : u1 = 8.0, m1 = 1.3, Translationon of similarity - center : c = 34, Elastic shear modulus : G = 1,800 kgf/cm 2 ,

280

11 Constitutive Equations of Soils

Fig. 11.16 Cyclic mobility of loose Niigata sand (after Ishihara et al., 1975). Test data and calculated results are depicted by the dashed and the solid lines, respectively.

Initial values:

Hardening function: F0 = 5.5 kgf/cm2 , Rotational hardening variable : β0 = 0, Center of similarity : s0 = −0.21 kgf/cm 2 , Stress : σ 0 = − 2.1 I kgf/cm2 ,

11.9 Simple Subloading Surface Model

281

11.9 Simple Subloading Surface Model The simulation of test data by the extended subloading surface model with the isotropic and anisotropic hardening is described above. In what follows, the simple initial subloading surface model for soils will be described below in order to exhibit differences from the other soil models. Now, by putting

s = α = 0, β = 0 , pe = 0, ξ = 0 , μ d = 0

(11.80)

in the afore-mentioned constitutive equations, the simple yield condition is obtained as follows:

⎫ ⎪ ⎪ ⎪ H dF F F (H ) = F0exp ρ γ , F ' ≡ dH = ρ − γ , ⎪⎪ − ⎬ ⎪ • p H = − t rD , h ( σ, H i , N) = − t r N, ⎪ ⎪ 2 6 sin φc ⎪ M= ⎪⎭ 3 − sinφc sin 3θ σ f (σ ) = p{1 + (

||σ ' || / p M

2

) },

(11.81)

for which the plastic modulus M p is given from Eq. (7.23) as − t r N U ( R) ) tr(Nσ) M p ≡( ρ −γ + R

(11.82)

and thus the plastic strain rate is described by p

D =

tr(N σ) N − t r N ( ρ − γ + U R( R) ) tr(Nσ)

(11.83)

and thus D = E−1 σ +

tr(N σ) N ( −ρt −r Nγ + U R( R) ) tr(Nσ)

(11.84)

Here, note that the subloading hardening with M p > 0 → tr(N σ) > 0 in Eq. (7.29) is induced over the critical state line fulfilling t r N = 0 . The partial derivatives of the yield function in Eq. (11.81) is given as follows:

η 2 ∂f ( p, η , M ) p η , ∂f ( p, η , M ) = 1+ (M ) , =2 M η p M ∂ ∂

282

11 Constitutive Equations of Soils

p η 2 ∂ f ( p, η , M ) = −2 ( M ) M ∂M

∂η = ∂ ||σ ' || / p = 1 { p + 1 ||σ || I } = 1 1 p2 τ 3 ' p (τ + 3η I ) ∂σ ∂σ

(11.85)

(11.86)

14 6 sin φ c M ∂M = − =− 8 + cos3θ σ ∂cos3θ σ (3 − sin φ c )(8 + cos3θ σ ) 2 =−

3 − sin φ c M2 14 6 sin φ c

(11.87)

∂τ = 1 ( I − τ2) ∂ σ ' || σ ' ||

(11.88)

∂cos3θ σ = 3 ( 6τ 2 − cos3θ σ τ) ∂σ' || σ'||

(11.89)

∂f (σ) ∂ f ( p, η , M ) = ∂σ ∂σ

η 2 p η 1 = − 1 {1 + ( ) } I + 2 (τ + 13η I ) M M M p 3

−2

p η 2 (− 3 − sin φc M 2) || σ3 || ( 6τ 2 − cos3θ σ τ) M (M ) ' 14 6 sin φ c

3 − sin φc η η 2 η = − 1 {1 − ( ) } I + 2 ( 6τ 2 − cos3θ σ τ) τ +3 M M M 3 7 6 sin φc M (11.90) On the other hand, the Cap model, which incorporates the Drucker-Prager model (Drucker and Prager, 1952) for the over-consolidated state into the Cam-clay model (Roscoe and Burland, 1968; Schofield and Wroth, 1968) for the normally-consolidated state, is most widely used for the prediction of soil deformation behavior. The predictions of the drained triaxial compression behavior of soils under the constant lateral stress by the subloading surface model in Eq. (11.84) and the Cap model are depicted in Fig. 11.17. Here, the curves of axial stress and volumetric strain vs. the axial strain in the loading from the heavily and the lightly over-consolidated states, i.e. points o and o' , respectively are shown in this figure. In the loading from the lightly over-consolidated state o' , the volumetric contraction proceeds and the axial stress increases up to the critical state. The abrupt transition from the elastic to the plastic state is predicted by the Cap model. On the other hand, the smooth behavior is predicted always by the subloading surface model as observed in experiments.

11.9 Simple Subloading Surface Model

283

Next, consider the loading from the heavily over-consolidated state o . In the subloading surface model, the first term − t r N/ (ρ − γ ) in the plastic modulus (11.82) decreases from the positive value to zero up to the point c on the critical state line (t r N = 0) . On the other hand, the second term U / R is always positive. Therefore, the plastic modulus keeps positive at the point c and thus the stress can go up the critical state line. After going up the critical state line, −t r N/ (ρ − γ ) tends to the negative value but U / R decreases so that the plastic modulus M p decreases to zero exhibiting the peak of stress when − t r N/ ( ρ − γ ) (< 0) and U/ R (>0) cancel mutually resulting in M p = 0 . Thereafter, − t r N/ (ρ − γ ) increases gradually and U/ R decreases more rapidly resulting in M p < 0 so that the stress decreases toward the critical state c .

trN > 0 q

•

q

y p

c−

c

c−

F<0

c'

y p c

y'

0

e lin te ta m

1 trN < 0 Dvp < 0

q

•

F >0

c'

y'

Normally-consolidate state

Stress path

Over-consolidate state

εa

−ε v

it Cr

Dvp > 0

ls ica

o

o

0 Fc' F0 ' Fc p Normal-yield surface Subloading surface at initial state at initial state Normal-yield surface Normal-yield surface at final state at final state

Dv / | Da | = max.

0

εa

−ε v

M

p

U ( R ) ( Nı ) tr R

M

Dvp

trN

o

B c Bp B c

p

−

εa

− t r N (Nı) ρ − γ tr

•

F

U M

εa U ( R ) ( Nı ) tr R

p

D

t r(N ı )

Stress-strain curve

½ model ¾ Elastoplastic state¿ Subloading surface

+

0

+

model u → ∞

+

+

−

max.

min.

0

+

−

−

Subloading surface model p

D =

D

tr(N ı ) N Mp

•

D p Dvp F U M tr( N ı )

trN

Elastic state

o'

B

− t r N (Nı) ρ − γ tr

−

c'

+ 0 (Critical state)

− t r N U ( R) M p ≡ ( ρ − γ + R ) tr(Nı ) ( tr( Nı ) ≥ 0)

0 (Critical state)

Fig. 11.17 Comparison of predictions of triaxial compression behavior under constant lateral stress by the Cap model and the subloading surface model

The sign of the plastic volumetric strain rate is identical to that of t rN , whilst the elastic volumetric strain is induced with the variation of pressure. The volumetric contraction is induced in the initial stage of loading. After going up the critical state line, the plastic volumetric expansion larger than the elastic volumetric contraction

284

11 Constitutive Equations of Soils

develops so that the volumetric expansion proceeds. However, reaching the peak stress tr(N σ ) = M p = Dve = 0 at which the subloading surface expands most and thus t r N becomes maximum, the peak of stress and the maximum ratio of volumetric

expansion strain rate vs. axial strain rate are induced simultaneously as has been found experimentally by Taylor (1948). Eventually, the fundamental deformation behavior in not only normally-consolidated but also over-consolidated states could be described concisely by the initial subloading surface model. The Cap model possesses various drawbacks compared with the subloading surface model. In other words, the subloading surface model is fundamentally modified from the Cap model as described in the following. 1) The Cap model falls within the framework of conventional plasticity with the yield surface enclosing the purely elastic domain. Therefore, the stress-strain curve is predicted to rise up steeply (elastically) to the peak stress, and subsequently it decreases suddenly exhibiting a softening. It leads especially to the unrealistic prediction of softening behavior. On the other hand, the subloading surface model can describe the realistic stress-strain curve with the smooth elastic-plastic transition since the plastic strain rate develops gradually as the stress approaches the normal-yield surface, i.e. the normal-yield ratio increases. 2) The Cap model necessitates the judgment of whether or not the stress lies on the yield surface, i.e. a yield judgment in addition to the judgment on the direction of strain rate in the loading criterion as shown in Eq. (6.68). On the other hand, the yield judgment is not required in the subloading surface model since the stress lies always on the subloading surface playing the role of the loading surface as shown in Eq. (7.27). 3) The Cap model requires the operation pulling the stress back to the yield surface, i.e. the return-mapping algorithm since the increments of stress or strain of finite magnitudes are input in the numerical calculation. On the other hand, the subloading surface model does not require it since it possesses an automatic controlling function to attract the stress to the normal-yield surface in the loading process as was described in 7.3. 4) The cap model additionally adopts the plastic potential surface of the conical shape having a dilatancy angle lower than the Drucker-Prager yield surface leading to the nonassociativity since an excessively large plastic volumetric strain is predicted if the associated flow rule is adopted and thus the constitutive equation becomes complicated, including additional material parameters. On the other hand, the subloading surface model can use the associated flow rule, whereas the outward-normal N of the subloading surface in the current stress is approximately identical to the outward-normal NDP of the plastic potential surface adopted in the Drucker-Prager model as shown in Fig. 11.18.

11.9 Simple Subloading Surface Model

285

DP

N

q Plastic potential surface assumed in Drucker-Prager model

N

ı

Drucker-Prager yield surface

0

it ic Cr

al

te sta

e lin

Normal-yield surface Subloading surface

p

Fig. 11.18 Outward-normal of subloading surface coinciding approximately with the plastic potential surface assumed in the Drucker-Prager model

5) The cap model adopts the nonassociativity for the Drucker-Prager yield surface. Therefore, it is accompanied with the asymmetry of the elastoplastic stiffness modulus tensor Mep . This fact engenders difficulty in the analysis of boundary value problems. On the other hand, the subloading surface model adopts the associativity leading to the symmetry of the elastoplastic stiffness modulus tensor. 6) The cap model predicts the failure surface describing the peak stresses determined uniquely by the Drucker-Prager yield surface itself, independent of the loading paths, because the interior of the yield surface is assumed to be a purely elastic domain. However, the surface depicted by connecting the peak stresses depends on the loading paths and exhibits nonlinearity with the increase of pressure in real soils on the contrary to the Coulomb-Mohr failure criterion. On the other hand, the subloading surface model can describe these facts (cf. Hashiguchi et al., 2002). 7) The cap model requires the tension cut for the Drucker-Prager yield surface, which runs out sharply into the negative pressure range. The subloading surface model does not require the tension cut because it adopts the normal-yield surface passing through the vicinity of the null stress state. 8) The cap model is accompanied with the singularity of the plastic potential in the intersecting lines of the Cam-clay, the Drucker-Prager, and the tension-cut yield surfaces, which would be unrealistic and would induce the difficulty of analysis of boundary-value problems. On the other hand, the subloading surface model adopts a single smooth normal-yield surface. Therefore, it does not induce the singularity of the plastic modulus. 9) The cap model predicts the simultaneous occurrence of the peak stress and the maximum volumetric compression in over-consolidated clays and dense sands, in contradiction to experimental fact. On the other hand, the subloading surface model provides the realistic prediction that the peak stress and the maximum ratio of volumetric expansion strain rate vs. axial strain rate occur simultaneously as described in the foregoing.

286

11 Constitutive Equations of Soils

10) The cap model requires at least two more material constants describing the inclinations of yield and plastic potential surfaces in addition to the material constants in the Cam-clay model. On the other hand, the subloading surface model requires only material constant u in the evolution rule of the normal-yield ratio despite the distinguishable high ability. In what follows, some comparisons of the simulations of typical triaxial test data by the Cap model and the subloading surface model are shown (Hashiguchi et al., 2002). The simulations of the test data measured by Skempton and Brown (1961) for Weald clay subjected to the drained triaxial compression with a constant lateral stress are shown in Fig. 11.19 where the material constants and the initial value are selected as follows:

ρ = 0.045, γ = 0.002, ν = 0.37, M = 1.2, M y = 0.574, M p = 0.071 for the Drucker-Prager model, u = 33.0 for the subloading surface model, F0 = 330 .0 kPa, whilst the initial stress state is σ0 = − 67.0I kPa . Here, the function M y and M p are the inclinations of yield and the plastic potential surfaces, respectively, of the Drucker-Prager model in the ( p, ||σ' ||) plane. The associated flow rule and the nonassociated flow rule are abbreviated as AFR and Non-AFR, respectively, in this figure. On the other hand, Eq. (7.17) is used for the evolution rule of the normal-yield ratio in the subloading surface model. The similar simulations for the test data of kaolinite-silt mixtures measured by Stark et al. (1994) are shown in Fig. 11.20 where the material constants and the initial value are selected as follows:

ρ = 0.1, γ = 0.006, ν = 0.3, M = 1.051, M y = 0.528, M p = 0.093 for the Drucker-Prager model, u = 35.0 for the subloading surface model, F0 = 6,000 .0 kPa, whilst the initial stress state is σ 0 = − 1275.0I kPa . The subloading surface model gives rise to the clearly better prediction than the Drucker-Prager model for both the axial stress-axial strain and the volumetric strain-axial strain curves. The curves predicted by the Drucker-Prager model are not smooth, which are formed by the three segments, i.e. the elastic, the elastoplastic and the critical state segments,

11.9 Simple Subloading Surface Model

287

whilst the former two form the concave curves of the ‘Eiffel-tower’ shape. Note that the adoption of the nonassociated flow rule in the Drucker-Prager model does not lead to the substantial improvement in simulation, whilst the subloading surface model adopting the associated flow rule gives the realistic prediction even for the volumetric strain. The parameter u is determined such that the stress-strain curve fit to the gentleness in the elastic-plastic transition.

200 Subloading surface model 150

||ı ' || (kPa)

100 Drucker-Prager model (Non-AFR) Drucker-Prager model (AFR) 50 : Test data 0 0

5

10

15

20

ε a (%) 2.0 Drucker-Prager model (AFR) 1.5 1.0

εv

0.5

Subloading surface model Drucker-Prager model (Non-AFR)

(%) 0.0

: Test data

0.5 1.0 0

5

10

15

20

ε a (%) Fig. 11.19 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Skempton and Brown, 1961) of Weald clay for the drained triaxial compression with the constant lateral pressure

288

11 Constitutive Equations of Soils 4000 Subloading surface model 3000

|| ı' || (kPa) 2000

Drucker-Prager model (Non-AFR) Drucker-Prager model (AFR)

1000

: Test data

0 0

5

10

15

20

25

ε a (%) 3 Drucker-Prager model (AFR)

2

εv

1

(%) 0 Subloading surface model Drucker-Prager model (Non-AFR)

1

: Test data 2 0

5

10

15

20

25

ε a (%) Fig. 11.20 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Stark et al., 1994) of kaolinite-silt mixtures for the drained triaxial compression with the constant lateral pressure

The simulations of the stress paths and the stress-strain curves to the test data measured by Bishop et al. (1965) for London clay subjected to the undrained triaxial compression are shown in Fig. 11.21 where the material constants and the initial value are selected as follows:

11.9 Simple Subloading Surface Model

289

1200 1000

|| ı' || (kPa)

Drucker-Prager model Yield surface in Drucker-Prager model

800

t ic C ri

a

ate l st

e lin

600 Test data p0 (kPa)

400

104 553

200 Subloading surface model 0 0

200

400

600

800

1000

1200

p (kPa) 1200

|| ı' || (kPa)

Test data p0 (kPa) 104 553

Drucker-Prager model ν =0.30 ν =0.45

1000 800 600 400 200

Subloading surface model 0

0

−1

−2

ε a (%)

−3

−4

−5

Fig. 11.21 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Bishop et al., 1965) for the undrained triaxial compression with the constant lateral pressure

ρ = 0.022, γ = 0.0063, M = 0.82, ν = 0.3 and ν = 0.45, M y = 0.62, M p = 0.21 for the Drucker-Prager model, ν = 0.3, u = 70.0 for the subloading surface model, F0 = 1,700.0 kPa.

290

11 Constitutive Equations of Soils

400 Drucker-Prager model Yield surface of Drucker-Prager model

300

it Cr

ica

l

te s ta

lin

e

|| ı' || (kPa)

200 Test data

p0 (kPa) 50 100 250

Subloading surface model

100

0

100

0

400

300

200

p (kPa)

400 Drucker-Prager model ν =0.30 ν =0.43 300

p0 (kPa)

||ı ' || (kPa)

Test data 200

50 100 250

100 Subloading surface model 0 0

−2

−4

ε a (%)

−6

−8

−10

Fig. 11.22 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Wesley, 1990) for the undrained triaxial compression with the constant lateral pressure

The similar simulations for the test data of red clay measured by Wesley (1990) are shown in Fig. 11.22 where the material constants and the initial value are selected as follows:

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

291

ρ = 0.035, γ = 0.012, M = 1.015, ν = 0.3 and ν = 0.43, M y = 0.767, M p = 0.24 for the Drucker-Prager model, ν = 0.3, u = 20.0 for the subloading surface model, F0 = 300.0 kPa. The test data are predicted fairly well by the subloading surface model. On the other hand, both the stress paths and the stress-strain curves predicted by the Drucker-Prager model are quite different from the test data, which are not smooth being formed by the three segments, where the Poisson’s ratio is selected two levels of ν = 0.30 and 0.45 in Fig. 11.21 and ν = 0.30 and 0.43 in Fig. 11.22. The simple subloading surface model has been widely applied to the analyses of soil deformation behavior (e.g. Topolnicki, 1990; Kohgo et al., 1993; Asaoka et al, 1997; Noda et al., 2000; Hashiguchi et al., 2002; Nakai and Hinokio, 2004; Wongsaroj et al., 2007).

11.10 Super-Yield Surface for Structured Soils in Natural Deposits Constitutive equations formulated to date are relevant to remolded soils. Asaoka et al. (2000, 2002) noticed that naturally deposited soils have structures and they have a strength greater than that of the remolded soils for same void ratio. To take account of this fact, they incorporated the novel surface, called the super-yield surface, which has a similar shape and orientation to the normal-yield surface but is larger than the normal-yield surface if soils have the structure. Constitutive equations based on this concept will be explained below in the revised forms. First, let the following super-yield surface be assumed, while it is postulated that the isotropic hardening is induced only by the plastic volumetric strain, independent of the deviatoric deformation. ˆ (H ) (11.91) f (σ )= RF where Rˆ ( ≥ 1) is the ratio of the size of the super-yield surface to that of the normal-yield surface and is called the structure ratio. Further, they incorporated the super-yield ratio R (0 ≤ R ≤ 1) describing the ratio of the size of the subloading surface to that of the super-yield surface, whilst the subloading surface can become larger than the normal-yield surface in structured soils. The subloading surface is described as ˆ (H ) (11.92) f (σ ) ( = RF ( H ))= RRF noting the normal-yield ratio R (0 ≤ R ≤ Rˆ ) is described by the product of structure ratio Rˆ and super-yield ratio R as

R = RRˆ

(11.93)

292

11 Constitutive Equations of Soils

q

Normal-yield surface

Subloading surface

f ( ı)= F (H )

f ( ı)= RF (H )

Super-yield surface

ˆ (H ) f ( ı)= RF

ı

F

0

RF

ˆ RF

p

Fig. 11.23 Super-yield, normal-yield and subloading surfaces

The super-yield, the normal-yield and the subloading surfaces are shown in Fig. 11.23. The time-differentiation of Eq. (11.92) leads to • • ˆ F + R Rˆ F + RRˆ F ′Hi t r ∂f (σ ) σ = R R ∂σ

(

)

(11.94) •

•

Now, assume for the upgradation ( Rˆ > 0) and the degradation ( Rˆ < 0) of structure as follows: •

•

1) The upgradation ( Rˆ > 0) or degradation ( Rˆ < 0) occurs only when the plastic strain rate is induced (D p ≠ 0) . •

2) The degradation of structure ( Rˆ < 0) proceeds when the deviatoric strain rate is induced (D p' ≠ 0) . •

3) The degradation of structure ( Rˆ < 0) proceeds when the plastic volumetric contraction ( Dvp < 0) is induced. The remolded state has the lowest grade of structure ( Rˆ = 1) , and thus the degradation occurs no more ( Rˆ ≥ 1) . •

4) The upgradation of structure ( Rˆ > 0) proceeds when the plastic volumetric expansion ( Dvp > 0) is induced. Based on these assumptions, let the following evolution rule of the structure ratio Rˆ be given as follows:

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

Uˆ d

Uˆ vc

Dvp < 0

p

D '≠0 0

1

Rˆ

0

1

293

e Uˆ v

Rˆ

Dvp > 0 0

1

Rˆ

Fig. 11.24 Functions Uˆ d, Uˆ d and Uˆ d in the evolution rule of Rˆ •

e p p c Rˆ = − Uˆ d ( Rˆ ) || D p' || − Uˆ v ( Rˆ ) 〈− Dv 〉 + Uˆ v ( Rˆ ) 〈Dv 〉

(11.95)

where Uˆ d , Uˆ vc, Uˆ ve are the functions of Rˆ fulfilling the conditions (Fig. 11.24).

⎧⎪ > 0 for Rˆ > 1 , ˆ c ˆ ⎧ ⎪> 0 for Rˆ > 1 , U v ( R )⎨ Uˆ d ( Rˆ )⎨ ⎪⎩= 0 for Rˆ = 1 ⎪⎩ = 0 for Rˆ = 1 Uˆ ve ( Rˆ ) > 0 for Rˆ ≥ 1

(11.96)

which are all the monotonically-increasing functions of Rˆ . The following simple equations are assumed hereinafter. m e e ξ Uˆ d ( Rˆ ) = uˆd ( Rˆ − 1) , Uˆ vc ( Rˆ ) = uˆ vc (Rˆ ζ − 1) , Uˆ v ( Rˆ ) = uˆ v Rˆ

(11.97)

where uˆd , m; uˆ vc, ζ ; uˆ ev, ξ ( ≤ 1) are the material constants. On the other hand, let the evolution rule of the super-yield ratio R be given by the following equation which is replaced the normal-yield ratio R to the super-yield ratio R in Eq. (7.13). •

R = U ( R ) || D p || for D p ≠ 0

(11.98)

where U is the monotonically increasing function of R fulfilling the conditions (Fig. 11.25):

⎧→ ∞ for R = 0, ⎪ U ⎨ = 0 for R = 1, ⎪ (< 0 for R > 1). ⎩

(11.99)

294

11 Constitutive Equations of Soils

• R = U ( R ) || D p ||

0

R

1

Fig. 11.25 Function

U in the evolution rule of the super-yield ratio R

The following simple equation is assumed putting Re = 0 in Eq. (7.15).

U ( R ) = u cot( π2 R )

(11.100)

where u is the material constant. The substitutions of Eqs. (11.38), (11.40), (11.95) and (11.98) into Eq. (11.94) yields p t r ∂f (σ ) σ = − RRˆ F' D v ∂σ

(

)

p ˆ e − R{Uˆ d ( Rˆ ) || D p' || + Uˆ vc ( Rˆ ) 〈− Dvp 〉 − Uˆ v ( Rˆ ) 〈Dvp 〉} F + U ( R) || D || RF

(11.101) which is rewritten as p RRˆ F tr( ) − ˆ t r( N σ ) N σ = RR F' D v

p ˆ e − R{Uˆ d ( Rˆ ) || D p' || + Uˆ vc ( Rˆ ) 〈− Dvp 〉 − Uˆ v ( Rˆ ) 〈Dvp 〉} F + U ( R) || D || RF

(11.102) noting the following relation due to Euler’s theorem of homogeneous function in degree-one. ∂f (σ ) = ∂σ

t r ( ∂f (σ ) σ) ˆ ∂σ N = RR F N t r(Nσ ) t r( Nσ )

(11.103)

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

295

Adopting the associated flow rule (7.19) in Eq. (11.102), one has

tr( N σ ) tr( N σ ) λ= , Dp = N Mp Mp

(11.104)

ˆ M p ≡ (− F' t r N+ U − U ) t r(Nσ) F R Rˆ

(11.105)

e c Uˆ ≡ Uˆ d || N' || + Uˆ v ( Rˆ ) 〈− t r N 〉 − Uˆ v ( Rˆ ) 〈t r N〉

(11.106)

where

setting

Only the softening and the hardening are predicted over and below, respectively, the critical state line in the Cam-clay model. On the other hand, the subloading hardening (tr( N σ ) > 0) can be also predicted over the critical state line because of the inclusion of the function U ( U in case of Eq. (11.105)) in the subloading surface model without the deviatoric hardening. Furthermore, both the subloading hardening (tr(N σ) > 0) and the subloading softening (t r( N σ ) < 0) can be predicted both over and below the critical state line because of the inclusion of the c e functions Uˆ d , Uˆ v , Uˆ v in addition to U in the subloading super-yield surface model (see Fig. 11.26). These facts are shown in Table 11.1. The undrained stress path in sands depicted in Fig. 11.6 and also the cyclic mobility, i.e. the

e lin te a t ls

q •

Dvp > 0 : F < 0

D

q

•

a itic Cr

Dvp > 0 : F < 0

a itic Cr D t r( N ı ) > 0 •

•

Dvp < 0 : F > 0 0

tr( N ı ) < 0

Dvp < 0 : F > 00 p

U − Uˆ > 0 § Structure ratio Rˆ : low, · ¨¨ Super - yield ratio R : low ¸¸ R Rˆ © ¹ Dense sand

0

a te l st

e lin

D

tr( N ı ) > 0

p

U − Uˆ < 0 § Structure ratio Rˆ : high, · ¨¨ Super - yield ratio R : high ¸¸ R Rˆ © ¹ Loose sand

Fig. 11.26 Subloading softening for a plastic volumetric expansion and subloading hardening for a plastic volumetric contraction

296

11 Constitutive Equations of Soils

Table 11.1 Signs of mechanical quantities above and below the critical state line in soil models

Cam-clay Model D

Subloading surface model D

Subloading superyield surface model D

M p , tr(N ı )

M p , tr(N ı )

M p , tr(N ı )

<0

< 0 or =0 or >0

< 0 or =0 or >0

=0

=0 or >0

< 0 or =0 or >0

>0

>0

< 0 or =0 or >0

Above C.S. line • trN > 0 , trD p > 0, F < 0 C.S. line • trN = 0 , trD p = 0, F = 0 Below C.S. line • trN < 0 , trD p <0, F > 0

butterfly-shape stress path after the decrease of pressure for the cyclic loading with the constant deviatoric stress amplitude under the undrained condition of sands could be described rigorously (Noda et al., 2007). Based on the concept of the extended subloading surface model described in Chapter 8, the above-mentioned subloading super-yield surface model is extended so as to describe the cyclic loading behavior as follows: Now, assume that the similarity-center s of the subloading and the super-yield surfaces moves with the plastic deformation (see Fig. 11.27). Then, the subloading surface is described by

ˆ ( H ) (= RF ( H )) f (σ, β) = RRF

(11.107)

where

σ ≡ σ −α ,

α = (1 − R)s

(11.108)

The material-time derivative of Eq. (11.107) leads to tr (

∂f (σ, β) ∂f ( , β) ∂f ( , β) σ ) − t r ( σ α ) + t r ( σ β) ∂σ ∂σ ∂β •

•

•

= R Rˆ F + R Rˆ F + RRˆ F ′H

(11.109)

where it holds from Eq. (11.108) that •

α = (1 − R ) s − R s

(11.110)

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

297

q Super-yield surface

ı Subloading surface

ı s

Į 0

p Normal-yield surface

Fig. 11.27 Normal-yield, subloading and super-yield surfaces

The similarity-center has to lie inside the super-yield surface in order that the subloading and the super-yield surfaces do not intersect mutually. Then, it has to be fulfilled that ˆ (H ) f (S, β ) ≤ RF

(11.111)

The time-differentiation of Eq. (11.111) at the limit state that s lies just on super-yield surface instead of the normal-yield surface (see Fig. 8.6) leads to

tr(

• • ∂f (s, β) ˆ (H ) s) + t r ( ∂f (s , β) β) − Rˆ F − Rˆ F ≤ 0 for f (s , β) = RF ∂s ∂β

(11.112) Noting the relation

tr(

∂f (s, β) ˆ , s} = RF t r{ ∂s

Eq. (11.112) becomes

• • ∂f (s, β) ∂f (s , β) ∂f (s, β) 1 s) + ˆ t r { ∂ s s}{t r ( ∂ β β) − Rˆ F − Rˆ F} ≤ 0 ∂s RF ˆ (H ) for f (s, β) = RF

i.e. •

tr

•

ˆ F ∂f (s , β) ∂f (s, H ) s +{ 1 t r ( β) − R − F }s β ˆ ∂s ∂ Rˆ RF

(

[

ˆ (H ) for f (s , β) = RF

]) ≤ 0 (11.113)

298

11 Constitutive Equations of Soils

Then, assume the following evolution rule of the similarity-center so as to fulfill the inequality (11.113). •

•

ˆ ∂f ( , β ) s = c|| D || σ +{ FF + Rˆ − 1 t r ( s β)}s ˆ R ∂β R RF

(11.114)

p

Substituting Eq. (11.114) into Eq. (11.110), it is obtained that •

•

• α = (1 − R)[c|| D || σ + { F + Rˆ − 1 t r ( ∂f (s , β ) β)}s ] − R s F Rˆ RF ˆ R ∂β p

(11.115) Further, the substitution of Eq. (11.115) into Eq. (11.109) leads to the following equation. t r(

∂f (σ , β) σ) ∂σ

−t r

(

{

+ t r(

•

•

∂f (σ , β) (1 − ) R ∂σ

∂f (s, β ) 1 [c||D p|| σR +{FF + RRˆˆ − RF tr( β)}s ] −R s }) ∂β ˆ •

• • • ∂f (σ , β) β) = R Rˆ F + R Rˆ F + RRˆ F β ∂

which, noting the relation

∂f (σ, β ) = ∂σ

tr

( ∂f (∂σσ, β ) σ) tr(Nσ )

N=

f (σ, β ) RRˆ F N= N tr(Nσ ) tr(Nσ )

(11.116)

becomes RRˆ F tr(N σ) tr( Nσ) •

•

• ˆ ˆ ∂f (s, H ) 1 − RR F tr N (1 − R) c|| D p || σ + { F + R − β)}s − R s tr( F ˆ ˆ R ∂β R RF tr(Nσ)

({

+ tr (

[

• • ∂f (σ, β) β) = R Rˆ F + R Rˆ F + RRˆ F• ∂β

]

})

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

299

i.e. tr(N σ) •

•

• ˆ ∂f (s, H) 1 − tr N (1 − R) c|| D p|| σ + { F + R − tr( β)}s −R s } F Rˆ RF ˆ R ∂β

[{

[

[ {RRˆ F 1

+ tr N

]

•

•

•

]

ˆ ∂f (σ, β) β) − R − R − F tr( β ∂ R Rˆ F

}σ ]= 0

i.e. •

•

• 1 1 F Rˆ p tr(N σ ) − tr N ( F + ˆ ){σ + (1 − R )s} + R ( R σ − s) + c ( − 1)|| D ||σ R R

([

∂f (σ, β) σ (1 − R) ∂f (s, H) s β) − − 1 tr ( tr( β) ˆ β ˆ ∂ ∂β F RR RF

]) = 0

(11.117)

Noting the relation

σ + (1 − R)s = σ − α + s − (s − α) = σ ⎫⎪ ⎬ 1 σ − s = 1 (σ + R s) − s = 1 σ ⎪ R

R

R

(11.118)

⎭

Eq. (11.117) becomes •

•

•

R 1 F Rˆ p tr(N σ ) − tr N (F + ˆ ) σ + σ + c ( − 1)|| D || σ R R R

([

−

1 tr( ∂f (σ, β) β) σ − (1 − R) t r ( ∂f (s, H) β) s ∂β ∂β RRˆ F Rˆ F

]) = 0

Substituting Eqs. (11.38), (11.95) and (11.98) into this equation, we have the consistency condition

( [{− FF′ D

tr(N σ) − tr N

p v

p p − 1 {Uˆ d || D p' || + Uˆ vc 〈− Dv 〉 − Uˆ ve 〈Dv 〉}}σ Rˆ

∂f ( , β) + {U + c ( 1 − 1)}|| D p|| σ − 1 tr( σ b (σ, H i )|| D p' ||)σ ˆ ∂β R R RR F ( − ) ∂f (s, H) σ − 1 R tr( b ( , H i )|| D p' || ) s ˆ β ∂ RF

]) = 0

300

11 Constitutive Equations of Soils

The substitution of the associated flow rule (8.25) to this equation leads to the following equation

( [{− FF′ trN

tr(N σ) − λ tr N

− 1 (Uˆ d || N' || + Uˆ vc 〈− tr N 〉 − Uˆ ve 〈 tr N〉 )}σ + { U + c ( 1 − 1)}σ Rˆ R R ∂f (σ, β) b (σ, H i )|| N' ||)σ − 1 tr( ˆ ∂β RR F

−

(1 − R ) ∂f ( s , H ) σ b ( , H i ) || N' ||) s tr( ˆ ∂β RF

]) = 0

from which the plastic modulus in Eq. (8.27) is derived as follows:

( [{− FF′ trN − R1ˆ {Uˆ

M p ≡ tr N

d || N' || + Uˆ vc 〈− tr N 〉

−Uˆ ve 〈 tr N〉} } σ +{U + c ( 1 − 1)}σ R R ∂f (σ , β) || || b (σ, H i ))σ − N' {tr ( ∂β RRˆ F ∂f (s, H) σ b ( , H i ))s} + R (1− R ) t r ( ∂β

])

(11.119)

In case of R = R , U ( R) = U ( R ) for soils without a structure, i.e. Rˆ = 1 leading to Uˆ d = Uˆ vc = Uˆ ve = 0 , the plastic modulus (11.119) reduces to Eq. (8.27). The structure ratio Rˆ is calculated by the numerical integration of Eq. (11.95) and R Rˆ can be calculated by the method described in 11.7 with the replacement of R to RRˆ . Then, the super-yield ratio R is calculated from them.

11.11 Numerical Analysis of Footing Settlement Problem

301

11.11 Numerical Analysis of Footing Settlement Problem Numerical analysis of footing settlement problem will be shown in this section (Mase and Hashiguchi, 2009). The prediction of peak load and post-peak behavior for the footing-settlement problem on sands having the high friction and dilatancy cannot be attained in fact by the usual implicit finite element method requiring the repeated calculations of total stiffness equation which needs quite large calculation time. On the other hand, it can be attained by the explicit dynamic relaxation method in which the dynamic equilibrium equation is solved directly without solving the total stiffness equation so that the calculation time is drastically reduced. The FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions; Cundall, P. and Board M., 1988; Itasca Consulting Group, 2006) based on the explicit dynamic relaxation method is adopted in the present analysis, in which the initial subloading surface model of soils with the automatic controlling function to attract the stress to the normal-yield surface is implemented as the constitutive equation. The calculation is executed by the Euler method without a calculation for convergence in this program by adopting small incremental steps so as not to influence on the calculation, while this fact is examined prior to the calculation. The finite elements are composed of eight-noded cuboidal elements. Each cuboidal element is divided into the two kinds of overlays, i.e. assembly of five tetrahedral sub-elements having different directions. Then, the deviatoric variables are analyzed using individual values in each tetrahedral sub-element. On the other hand, isotropic variables are analyzed using averaging values in ten tetrahedral sub-elements in order to avoid the over-constraint problems common in finite element calculations for dilatant materials, i.e. the dilatancy locking.

Test data used for numerical simulation The test data of footing settlement phenomenon on sand layers under the plane strain condition are used for the present analysis. The sizes of the test apparatus of type A (Tatsuoka et al., 1984) and type B (Tani, 1986) have the same height 49cm and depth 40cm and the different widths 122cm and 183cm, respectively. The size of type C (Okahara et al., 1989) has the height 400cm and depth 350cm and the widths 7002cm. The footings width, denoted as B, is taken 10 cm for the types A and B and 50cm for the type C. The sand layers has been prepared carefully by the air-pluviation method for the dried Toyoura sand in order to obtain the same homogeneous layers but the test data exhibit dispersion more or less test by test despite of the laborious preparation work.

Numerical analysis and comparison with test data The finite element meshes in the present analyses for the simulations of the test data are shown in Fig. 11.28. The nodal points of soil layer contacting with the footing

302

11 Constitutive Equations of Soils

and the bottom of soil bin are fixed to them, respectively. On the other hand, the nodal points at the side walls can move freely in the vertical direction. The right half of soil layer is analyzed in order to reduce the calculation time as has been done widely even for searching the localized deformation (cf. e.g. Sloan and Randolph; 1982; Pietruszczak and Niu; 1993; Stallebrass et al., 1997; Borja and Tamagnini, 1998; Siddiquee et al., 1999; de Borst and Groen, 1999; Sheng et al., 2000; Borja et al., 2003). First, the analysis of deformation caused by the gravity force was performed. Then, the vertical displacement of footing is given by incremental steps of 10−5 ~ 5 × 10−4 cm. The material parameters in the subloading surface model are selected as

F0 = 350 kPa, φc = 30 , ξ = 0.001 ,

ρ = 0.0015, γ = 0.00015 , pe = 0.01kPa, ν = 0.3 ,

φd = 29 , μd = 0 .2 , u = 15.0 where Eq. (7.17) is used for the evolution rule of the normal-yield ratio in the subloading surface model. The values of material parameters listed above are used for all the following numerical calculations because Toyoura sands having the same initial void ratio 0.66 are used in these tests. The comparisons of test and calculated results are shown in Fig. 11.29, where the prediction by Siddiquee et al. (1999) is also depicted in (c). In this figure qm is the average footing pressure, γd is the unit dry weight, Nγ is the normalized footing pressure and S is the settlement. The qualitative trends of test results and the quantitative simulation to some extent are captured and the ultimate loads, i.e. bearing capacities are predicted well by the present analyses, although the analyses are performed for the sand with the high friction and dilatancy. Here, the post-peak behavior, i.e. the increase of load after exhibiting once the minimal value is also predicted well qualitatively. It would be provided by the adoption of the up-dated Lagrangian calculation realizing the accumulation of displacements by updating the positions of nodal points, which results in the upsurge of soils around the footing and thus the increase of footing load. However, the quantitative prediction of post-peak behavior would require the further study taking account of the tangential inelastic strain rate due to the stress rate tangential to the loading surface (Hashiguchi and Tsutsumi, 2001) and the gradient effect (cf. Hashiguchi and Tsutsumi, 2006) by introducing the shear-embedded model (cf. Pietruszczak and Mroz, 1981; Tanaka and kawamoto, 1988) for example, which will be described in 13.3.

11.11 Numerical Analysis of Footing Settlement Problem

C.L. 56cm

49cm

5cm

(a) Type A ( B : 10cm)

C.L. 86cm

49cm

5cm

(b) Type B (B : 10cm) 325cm

400cm

C.L. 25cm

(c) Type C (B : 50cm)

Fig. 11.28 Finite element meshes

303

11 Constitutive Equations of Soils

300

Normalized footing pressure Nγ =2q γdΒ

Normalized footing pressure Nγ =2q γdΒ

304

FDM by subloading surface model

Test (Tatsuoka et al., 1984): e=0.66 Test (Tatsuoka et al., 1984): e=0.66

200

100

0

0.00

0.05

0.10

Relative settlement

0.15

Normalized footing pressure Nγ =2q γdΒ

FDM by subloading surface model Test (Tani, 1986): e=0.669,BC2 Test (Tani, 1986): e=0.669,BC3

200

100

0

0.00

0.05

0.10

Relative settlement

S B

(a) Type A (B: 10cm)

0.15 S B

(b) Type B (B: 10cm)

300 FDM by subloading surface model

qm : Average footing pressure

FEM (Siddiquee et al., 1999)

γd : Unit dry weight

Test (Okahara et al., 1989): e=0.66

S : Settlement B: Footing width

Test (Okahara et al., 1989): e=0.66

200

300

100

0

0.00

0.05

0.10

Relative settlement

0.15 S B

(c) Type C (B: 50cm)

Fig. 11.29 Comparisons of test and calculated results for footing settlement phenomenon

The displacements of nodal points from the initiation of settlement are shown in Fig. 11.30 at the settlement 11mm，15mm，80mm for Type A, B and C, respectively, which are the final stage of calculation. The Prantdl’s slip line solution with the triangle wedge, the logarithmic spiral zone and the passive Rankine zone is observed clearly in this figure. On the other hand, the soils in the periphery of footing inevitably experience the null or further negative pressure since they are pulled into the vertical direction as the footing settlement proceeds (see Fig. 11.31). It causes the singularity of plastic modulus for the normal-yield surface passing through the origin of stress space at which the normal-yield and the subloading surfaces contact with each other. This defect is improved in the present model by making the normal-yield surface translate to the region of negative pressure as shown in Fig. 11.5, whilst the numerical difficulty can be avoided although the translation was taken quite small

11.11 Numerical Analysis of Footing Settlement Problem

Displacement (cm) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(a) Type A (B : 10cm)

Displacement (cm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.5

I

(b) Type B (B : 10cm)

Displacement (cm) 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

(c) Type C (B : 50cm)

Fig. 11.30 Deformed finite element meshes at final step

305

306

11 Constitutive Equations of Soils

as 1/1000 in size of the normal-yield surface. In addition, the impertinence that the volume becomes infinite elastically is avoided by shifting the isotropic consolidation characteristic into the negative range of pressure as shown in Fig. 11.1. It should be emphasized that the stable analysis cannot be executed without these improvements.

C.L.

p (kPa) −0.0001 0 10 10 20 20 30 30 40 40 50 50 60 60

0

I t

Fig. 11.31 Distribution of mean pressure for type A at final step

The pertinent result for the footing-settlement problem on the sand with a high friction, one of the difficult problems in soil mechanics, is obtained in the present study as described above. Here, the peak, the subsequent reduction and the final increase of footing load are predicted well qualitatively and quantitatively to some extent. The reasons for succession are summarized as follows: 1. The subloading surface model applied in the present analysis has the advantages: i) It is furnished with the automatic controlling function to attract the stress to the yield surface, whilst all other elastoplastic constitutive models are required to incorporate a return-mapping algorithm to pull back the stress to the yield surface in the plastic deformation process in the normal-yield state. The distinguished advantage enables us to execute an accurate calculation by the program FLAC3D adopting the simple Euler method without a convergence calculation process. ii) It is not required the judgment whether or not the yield condition is fulfilled in the loading criterion. iii) It adopts the

11.11 Numerical Analysis of Footing Settlement Problem

307

associativity of the flow rule and thus it leads to the symmetry of elastoplasic constitutive matrix. Therefore, this model possesses the distinguishable adaptability in numerical calculation. 2. The subloading surface model of soils applied in the present analysis has the advantages: i) It is capable of describing the softening behavior and dilatancy characteristics quite realistically, predicting the simultaneous occurrence of the peak load and the highest dilatancy rate as has been found experimentally by Taylor (1948). ii) It has the full regularity since the normal-yield surface does not pass through the zero stress point and thus the subloading surface is always determined uniquely. In addition, the elastic property is improved such that the elastic bulk modulus does not become zero for the stress inside the normal-yield surface. 3. The finite difference program FLAC3D adopted in the present study is based on the explicit-relaxation method which enables us to shorten the calculation time drastically since it is not required to solve the total stiffness matrix.

Chapter 12

Corotational Rate Tensor 12 Corotational Rate Te nsor

It was studied in Chapter 4 that the material-time derivatives of state variables, e.g. stress and internal variables in elastoplasticity do not possess the objectivity and thus, instead of them, we must use their corotational derivatives. This chapter focuses on the responses of simple constitutive equations introducing corotational rates with various spins including the plastic spin.

12.1 Hypoelasticity Consider the following hypo-elastic constitutive equation in Eq. (5.22).

σD = λ (trD)I + 2GD where the symbol

(12.1)

L is replaced to λ . Equation (12.1) reduces to the following

equation noting that σ12 = σ 21 , ω 12 = −ω 21 and using Eqs. (2.144) and (4.33) for the simple shear described in 2.4.2.

⎡σ• 11 − 2σ ω σ• 12 + (σ − σ )ω ⎤ 12 12 11 22 12 ⎢ ⎥ = G ⎡0 ⎢1 ⎢ ⎥ ⎣ Sym. σ• 22 + 2σ12 ω12 ⎦⎥ ⎣⎢

1⎤ • γ 0 ⎥⎦

(12.2)

12.1.1 Jaumann Rate When the Jaumann rate is adopted for the corotational rate, Eq. (12.2) leads to the following equation by setting ω = W with Eq. (2.144).

⎡ ⎢σ• 11 − γ• σ 12 ⎢ ⎢ ⎢⎣ sym.

⎤ γ σ• 12 + (σ 11 − σ 22 ) ⎥ •

2

•

σ• 22 + γ σ 12

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 309–325. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

⎡0 ⎥ = G⎢ ⎣1 ⎥ ⎥⎦

1⎤ • γ 0 ⎥⎦

(12.3)

310

12 Corotational Rate Tensor

from which we have ⎫ ⎪ ⎪ • γ •⎪ σ• 12 + (σ 11 − σ 22 ) = G γ ⎬ 2 ⎪ • ⎪ • σ 22 + γ σ 12 = 0 ⎪ ⎭ •

σ• 11 − γ σ 12 =0

(12.4)

Substituting • σ 22 = −σ 11 , γ• = σ 11 σ12

(12.5)

obtained from the first and the third equations into the second equation in Eq. (12.4), yields • σ• 11 11 σ σ• 12 + σ G = 11 σ σ 12

(12.6)

12

the time-integration of which is given as

σ 12 = 2Gσ 11 − σ 112

(12.7)

Substituting this equation into the second equation of Eq. (12.4), we have σ• 11 G

•

σ 11

1 − (1 − G

2

=γ

)

the integration of which is given by σ11

cos−1 (1 − G

)=γ

i.e.

σ11 = −σ 22 = G(1 − cosγ )

(12.8)

The substitution of Eq. (12.8) into Eq. (12.7) leads to

σ 12 = Gsin γ

(12.9)

The continuum spin W designates the instantaneous rate of rotation of the principal directions of strain rate, i.e. the instantaneous rate of rotation of the cross depicted momentarily on the material surface as described in 2.3. Therefore, if it is

12.1 Hypoelasticity

311

2.8

σ12 /G

2.4 2.0

Green-Naghdi rate with ȍ R

1.6 1.2 1.0 0.8

Jaumann rate with W

0.4 0.0

π /2

π

γ

− 0.4 − 0.8

Fig. 12.1 Description of simple shear deformation of hypoelastic material by Jaumann rate and Green Naghdi rate (Dienes, 1979)

used in the simple shear deformation with the constant shear strain rate, i.e. γ• = const. leading to W = const. , the material is regarded to rotate in a constant velocity, while the strain rate D is also kept constant. Then, when the material rotates by 180 D , the material becomes to be subjected to the shear strain rate in the opposite direction. Eventually, the shear stress oscillates in the sine curve for the shear strain as seen in Fig. 12.1 calculated by Eq. (12.9) (cf. e.g. Dienes, 1979). 12.1.2 Green-Naghdi Rate When one adopts the Green-Naghdi (Dienes) rate for the corotational rate with the relative spin ω = Ω R , i.e. Eq. (4.43), noting

D=G

2 cos 2θ

⎡0 1⎤ • ⎢ ⎥θ , ⎣1 0 ⎦

• ⎡0 1 ⎤ • Ω R = R RT = ⎢ ⎥θ ⎣ −1 0 ⎦

(= 4 +2γ

2

⎡0 1 ⎤ • ⎢ ⎥γ ⎣ −1 0 ⎦

) (12.10)

312

12 Corotational Rate Tensor

Eq. (12.2) reads • ⎡• ⎢σ 11 − 2σ12 θ ⎢ Sym. ⎣⎢

σ• 12 + (σ 11 − σ22 ) θ ⎥⎤ •

• ⎥ σ• 22 + 2σ12 θ ⎦⎥

=G

2 cos 2θ

⎡0 ⎢1 ⎣

1⎤ • θ 0 ⎥⎦

(12.11)

from which we have ⎫ ⎪ • •⎪ • ⎪ 2 σ 12 + (σ 11 − σ22 ) θ = G θ⎬ cos 2θ ⎪ • ⎪ σ• 22 + 2σ12 θ = 0 ⎪⎭ •

σ• 11 − 2σ12 θ =0

(12.12)

It is obtained that •

σ 22 = − σ 11 θ• = 1 σ 11 2 σ 12

，

(12.13)

from the first and the third equations, and

dσ 12 = 1 d 2σ 11 dθ 2 dθ 2

(12.14)

from the first equation in Eq. (12.12). Substituting Eqs. (12.13) and (12.4) into the second equation in Eq. (12.12), we have the ordinary differential equation d 2σ 11 + 4σ = 4G 11 dθ 2 cos2θ

(12.15)

The roots of the characteristic equation of the second-order homogeneous linear differential equation for Eq. (12.15) are given by m 2 + 4m = 0 → m =

± −16 ± 2i 2 =

Thus, the complementary function of Eq. (12.15) is given by the following equation.

σ11 = Acos2θ + Bsin2θ = A2 + B 2 sin{2θ + tan( A / B)}

(12.16)

where A, B are the integral constants. Further, adding the particular solution for Eq. (12.15) itself, the general solution of Eq. (12.15) is obtained as follows:

12.2 Kinematic Hardening Material

313

σ 11 = A2 + B 2 sin(2θ + tan( A / B )) + 4G (cos2θ ln cosθ + θ sin2θ − sin2θ ) (12.17) Assuming that the initial stress is zero, Eq. (12.17) becomes

σ 11 = 4G (cos2θ ln cosθ + θ sin2θ − sin2θ )

(12.18)

Furthermore, substituting Eq. (12.18) into Eq. (12.14), we have

σ12 = 1 dσ11 =2Gcos2θ (2θ − 2 tan θ ln cosθ − tanθ ) 2 dθ

(12.19)

These equations have been derived by Dienes (1979). The relative spin Ω R designates the mean rate of rotation of the cross depicted on the material surface at the beginning of deformation. Therefore, it coincides with the continuum spin W at the initial state but it decreases gradually with the shear deformation. Then, the oscillation of shear stress observed in Jaumann rate is not predicted if the Green-Naghdi rate is adopted as the corotational rate as seen in Fig. 12.1 calculated by Eq. (12.19).

12.2 Kinematic Hardening Material For the sake of simplicity, consider the response of a rigid plastic material fulfilling D = D p and assume the linear kinematic hardening in Eq. (10.4), i.e.

αD = 23 h a Nˆ || D p || = 23 h a D p

(12.20)

while it is set that a p = (2 / 3)h a in accordance with Dafailas (1985). Then, it holds for the simple shear that ⎡αD αD ⎤ ⎢ 11 12 ⎥ = 1 h a ⎡0 1⎤ γ• ⎢ ⎥ ⎢D D ⎥ 3 ⎣1 0 ⎦ α α 22 21 ⎣ ⎦

(12.21)

by substituting Eq. (2.144) into Eq. (12.20). Further, substituting Eqs. (2.144), (2.164) and (4.33) into Eq. (12.21), we have

314

12 Corotational Rate Tensor • ⎡α• − 2α ω α 12 + (α11 − α22 )ω 12⎤ 1 ⎡0 12 12 ⎢ 11 ⎥ = ha ⎢ ⎥ 3 ⎢⎣1 • Sym. α 22 + 2α12 ω12 ⎦⎥ ⎣⎢

1⎤ • γ 0 ⎥⎦

(12.22)

where

ω = ⎡⎢

0 1⎤ • ⎥ z (γ ) γ − 1 0 ⎣ ⎦

(12.23)

1/2 for ω = W ⎧ z (γ ) = ⎨ 2 p γ ⎩2/(4 + ) for ω = Ω

(12.24)

The substitution of Eq. (12.23) into Eq. (12.22) leads to • ⎡α• − 2 α 12 z (γ ) γ ⎢ 11 ⎢ Sym. ⎣⎢

α 12 + (α11 − α22 ) z (γ ) γ• ⎤ 1 ⎡ 0 1⎤ • ⎥ = ha γ (12.25) ⎥ 3 ⎢⎣ −1 0 ⎥⎦ • • α + α 22 2 12 z (γ ) γ ⎦⎥ •

from which we have ⎫ ⎪ ⎪ α• 12 + (α11 − α22 ) z (γ ) γ• = 1 h a γ• ⎬ 3 ⎪ • • ⎪ α 22 + 2α12 z (γ ) γ = 0 ⎭

α• 11 − 2 α12 z (γ ) γ• =0

(12.26)

In addition, noting α11 = − α22 , we have

α11' = − α 22' =2 z (γ )α12 ⎫ ⎪ ⎬

(12.27)

α 12' + 2z (γ )α11 = 1 h a ⎪ 3

⎭

where ( )' = d ( )/d γ . Differentiating Eq. (12.27), we have

α11 '' − 2zα12' − 2α12 z' = 0 ⎫⎪ ⎬

α''12 + 2α11' z + 2α11z' = 0 ⎭⎪

(12.28)

12.2 Kinematic Hardening Material

315

which noting Eq. (12.27), becomes ⎫

α11 '' − 2z(1 h a − 2zα11 ) − 2 α11' z' =0 ⎪

⎪ ⎬ α''12 + 4 zα12 z + 2 1 (1 h a − α 12' )z' = 0 ⎪ ⎪⎭ 2z 3 2z

3

Then, it is obtained that ⎫

α11 '' − zz' α11' + 4 z 2α11 − 2 h a z = 0 ⎪ ⎪ ⎬ ⎪ 1 z z α''12 − z' α 12' + 4 z 2α12 + h a z' = 0 ⎪ 3 ⎭

(12.29)

3

12.2.1 Jaumann Rate

The substitution of Eq. (12.24) 1 into Eq. (12.29) leads to

α11 '' + α11 − 12 ap = 0⎫⎪

(12.30)

⎬ ⎪ ⎭

α 12 '' + α12 = 0

from which, noting the initial condition α11 = 0, α12 = 0 for γ = 0 , we have

α 11 = −α 22 = 1 ap (1− cos γ ) ⎫⎪ ⎪ ⎬ ⎪ ⎪⎭

2

α 12 = 1 ap sinγ 2

(12.31)

It is obtained from Eq. (12.31) that

σ 11 = α 11 = − σ 22 = − α 22 = 1 ap (1− cosγ ) ⎫⎪ 2 ⎪ σ 12 = 1 F + α 12 = 1 F + 1 ap sin γ 3

3

2

⎬ ⎪ ⎪⎭

Both σ 11 and σ 12 oscillates in sine curves as shown in Fig. 12.2.

(12.32)

316

12 Corotational Rate Tensor 4

σ 12

Green-Naghdi rate with ȍ R

3

3

σ 11 2

2 Jaumann rate with W

1

1

0 0

2

4

6 (a )

8

0

10

γ

0

2

4

6 (b)

8

γ

10

3

α 12 2

1

0

−1

0

1

2

(c)

γ

3

Fig. 12.2 Description of simple shear deformation of kinematic hardening material by Jaumann and Green-Naghdi rates (Dafalias, 1985)

12.2.2 Green-Naghdi Rate

Substituting Eq. (12.24) 2 and

−4γ z ' = ( γ 2 )2 4+

into Eq. (12.29), we have ⎫ ⎪ ⎪ ⎪ ⎪⎪ ⎬ ⎪ −4γ −4γ 2 )2 2 )2 ⎪ γ γ ( ( 1 4 4 + + 4 α + ap α''12 − α 12' + 4 = 0⎪ (4 + γ 2 ) 2 12 2 2 2 ⎪ 4+γ 2 4+γ 2 ⎪⎭

−4γ γ 2 )2 α ( 4 α11 '' − + ' + 4 ( 4γ 2 ) 2 α11 − ap 2γ 2 = 0 11 2 4+ 4+ 4+γ 2

(12.33)

12.3 Plastic Spin

317

that is to say, we obtain

α11 '' + 2γγ 2 α11' + 16 2 2 α11 − 12 ap 4γ 2 = 0 4+ 4+ (4 + γ )

⎫ ⎪ ⎪ ⎬ γ α''12 + 2γ 2 α 12' + 16 2 α12 − 1 ap 2 2h a = 0 ⎪ 2 4+γ 4+γ (4 + γ 2 ) ⎭⎪

(12.34)

the general solution of which is derived as the following equation by the method of variable coefficients (Dafalias, 1983). 2 1 [4γ {4 tan −1 ( γ ) − γ } − 4(γ 2 − 4) ln ] 2 4+γ 2 4+γ 2

⎫ ⎪ ⎪ ⎬ 2 γ 1 3 1 − 1 ⎪ 2 − 4)4 tan ( ) − 4γ (1 + 4 ln γ 4( [ )] α12 = ap γ − 2 2 4+γ 2 4 + γ 2 ⎪⎭

α11 = 1 ap 2

(12.35) The relation of σ 11 , σ 12 to α11 , α12 is given by Eq. (12.32) also in this case. A oscillation is not predicted in the simple shear deformation as shown in Fig. 12.2.

12.3 Plastic Spin The above-mentioned Jaumann rate and the Green-Naghdi rate do not depend on the substructure of material, rather are uniquely determined only by the external appearance of the material. However, the mechanically meaningful rotation would be the spin of substructure, as known presuming the crystals of metals or the annual ring of woods, which would be the rotation of the principal direction of anisotropy (Kroner, 1960; Kratochvil, 1971; Mandel, 1971). The concept of the plastic spin is proposed in order to incorporate such rotation into elastoplastic constitutive equations (Dafalias, 1983, 1985; Loret, 1983). In what follows, in order to interpret the mechanical meaning of the plastic spin, assume the rigid plasticity and the simplest anisotropy, i.e. the traverse anisotropic material (Fig. 12.3) with the parallel line-elements of substructure having the direction eˆ 1 inclined π / 4 from the fixed base e1 in the initial state of deformation and rotates the angle ϕ in the clockwise direction with the increase of shear strain (Dafalias, 1984). Here, Eq. (6.27) is postulated for the multiplicative decomposition. Then, it holds that ⎧cos(π / 4 − ϕ ) ⎫ • ⎧ sin(π / 4 − ϕ ) ⎫ • ⎬ , eˆ 1 = ⎨ ⎬ϕ ⎩sin(π / 4 − ϕ ) ⎭ ⎩− cos(π / 4 − ϕ ) ⎭

eˆ1 = ⎨

(12.36)

318

12 Corotational Rate Tensor

while Ve = I (rigid plasticity) leading to We = 0 and it holds that

p = W p . Here, W

referring Fig. 12.1,

tan(π / 4 − ϕ ) = 1 γ 1+

(12.37)

from which one has − ϕ• − γ• • = = − γ tan 2 (π / 4 − ϕ ) cos 2 (π / 4 − ϕ ) (1 + γ ) 2

(12.38)

Then it holds that • γ• γ• ϕ• = γ sin 2 (π / 4 − ϕ ) (= [1 − cos{2(π / 4 − ϕ )}]) = (1 − sin 2ϕ )

2

2 (12.39)

Using Eq. (12.39) along with Eq. (12.36), it is obtained that •

⎧ sin(π / 4 − ϕ ) ⎫ γ• ⎬ (1 − sin 2ϕ ) 2 ⎩ − cos(π / 4 − ϕ ) ⎭

eˆ 1 = ⎨

(12.40)

The substructure spin is given by the continuum spin W in the initial state of deformation but it becomes smaller than W with shear deformation. Then, let the decrease of spin from the continuum spin be called the plastic spin, and let it be ˆ p . Referring to the strain rate circle in Fig. 12.1 based on Fig. 2.5 in denoted by W ˆ p is given as 2.3, W

sin 2ϕ ⎤ • ˆ p ≡1⎡ 0 γ W 2 ⎢⎣− sin 2ϕ 0 ⎥⎦

(12.41)

Here, it holds from Eqs. (2.144) and (12.41) that

⎡ W12 − Wˆ12p ⎤ ⎧cos(π / 4 − ϕ ) ⎫ 0 ⎢ ⎥⎨ ⎬ ˆp 0 ⎣⎢ −(W12 − W12 ) ⎦⎥ ⎩sin(π / 4 − ϕ ) ⎭ • ⎡ ⎤ 0 ( γ /2)(1 − sin 2ϕ ) ⎥ ⎧cos(π / 4 − ϕ ) ⎫ =⎢ ⎨ ⎬ ⎢ • ⎥ ⎩sin(π / 4 − ϕ ) ⎭ 0 ⎣ − ( γ /2)(1 − sin 2ϕ ) ⎦

⎧ sin(π / 4 − ϕ ) ⎫ γ• =⎨ ⎬ (1 − sin 2ϕ ) 2 ⎩− cos(π / 4 − ϕ ) ⎭

12.3 Plastic Spin

319

tan −1γ ϕ

a

e2

π /4

eˆ 2

eˆ1 e1

aγ

a ω (= W12 + D 12 )

W12

2ϕ •

γ /2

0

Wˆ12p

Dn

Fig. 12.3 Substructure spin

Noting Eqs. (12.36), (12.40) and (12.41) in this equation, one obtains the following expression for the rotation of substructure. •

ωeˆ1 = eˆ1

(12.42)

where ˆp ω≡ W−W

(12.43)

Next, consider the same problem by the deformation of crystals of metals (Kuroda, 1996). If the substructure does not rotate, it holds for the slip system in Fig. 12.4 that

v = γ• (x • n)s = γ• (s ⊗ x)n , vi = γ• ( xr nr ) si

(12.44)

320

12 Corotational Rate Tensor • ∂v • • Lp = ∂v = γ s ⊗ n , Lijp = i = γ δjr nr si = γ si nj ∂x ∂xj

(12.45)

v ∂v 1• D p = 2 γ (s ⊗ n + n ⊗ s) , Dijp = 1 ( ∂ i + j ) = 1 γ• (si nj + ni sj ) 2 ∂xj 2 ∂xi

(12.46) 1• v v ∂ j • ∂ p 1 1 i W = 2 γ (s ⊗ n − n ⊗ s) , Wij = 2 ( x − x ) = 2 γ (si nj − ni sj ) ∂ j ∂ i (12.47) p

Eqs. (12.44)-(12.47) are extended for multi slip systems as follows: N

• (α ) (α )

Lp = ∑ γ

s

⊗ n(α )

(12.48)

α =1

• (α )

N

Dp = ∑γ α =1

{12 (s α

( )

N

• (α )

⊗ n(α ) + n(α ) ⊗ s(α ) )} = ∑ γ

p (α )

p ( α ) ≡ 1 (s ( α ) ⊗ n ( α ) + n ( α ) ⊗ s ( α ) ) 2 N

• (α )

Wp = ∑ 2γ 1

α =1

(12.49)

α =1

(12.50) N

• (α )

(s(α ) ⊗ n(α ) − n(α ) ⊗ s(α ) ) = ∑ γ

w (α ) (12.51)

α =1

) ) w (α ) ≡ 1 ( s ( α ) ⊗ n ( α − n (α ⊗ s (α ) ) 2

(12.52)

ˆ p. Needless to say, it holds in this formulation that ω = 0, W = W p = W The simple example of the plastic spin is shown above. Dafalias (1985) provided the general mechanical interpretation of the plastic spin based on the multiplicative decomposition with the postulate of Eq. (6.27) as follows.

v

γ

τ

.

γ .

x

n

τ

s

Fig. 12.4 Slip system and slip deformation (Kuroda, 1996)

12.3 Plastic Spin

321

The velocity gradient L in Eq. (6.9) can be additively decomposed into the three parts, i.e. the rigid body spin, the elastic deformation rate and the plastic deformation rate as follows: D p −1 −1 D e −1 L = ω + V V e + Ve F F p Ve

(12.53)

De • e Dp • p V = V − ωVe + V eω, F = F − ωF p

(12.54)

where

D e and D p are the corotational rates of e and p , respectively. Here, V F V F The strain rate D and the continuum spin W are decomposed by Eq. (12.53) as follows: D e e −1 ) + ( e D p p −1 e −1)s D = (L ) s = ( V V s V F F V −1 D p −1 −1 D W = (L ) a = ω + ( V e V e ) a + ( V e F F p V e ) a

(12.55)

(12.56)

Here, using Eq. (6.27), i.e. and assuming that the elastic deformation is infinitesimal and the unloading process is not accompanied with a rotation, which leads to FD e = VD e = D e, (FD e F e −1 )a ≅ 0 , Eqs. (12.55) and (12.56) reduce to D = De + D p

(12.57)

ˆ p W =ω+W

(12.58)

where D p p −1 ) ˆ p ≡ (F W F a

(12.59)

Although the corotational spin ω is expressed formally as the subtraction of the ˆ p from the continuum spin W in Eqs. (12.43) and (12.58), it means plastic spin W the rigid-body spin of substructure but, needless to say, it is impertinent to be called an elastic spin which is merely ignored in (12.56). Substituting Eq. (12.58) and the Jaumann rate (4.41) into Eqs. (4.33), the corotational rate is given by : D • ˆ p )T + T( W − W ˆ p ) = T +W ˆ pT −TW ˆ p T = T −(W −W

(12.60)

Noting that the plastic spin is the skew-symmetric tensor, the following explicit

．

equation is proposed by Zbib and Aifantis (1989)

ˆ p = ρ (σD p − D p σ ) W 2

where ρ is the function of internal variables Hi .

(12.61)

322

12 Corotational Rate Tensor

The relation of the corotational rate and the Jaumann rate of Cauchy stress is given from Eq. (12.60) with Eq. (12.61) as

ˆ pσ − σW ˆ p σD = σ + W :

: ˆ −N ˆ σ)σ − σ (σN ˆ −N ˆ σ)} = σ + (ρ /2)λˆ{(σ N

i.e.

σD = σ + λˆσ N :

(12.62)

where

σ N ≡ (ρ /2)(2 σNˆ σ − Nˆ σ 2 − σ 2Nˆ )

(12.63)

Substituting Eq. (12.62) into Eq. (6.94), it follows that

ˆ (σ + λˆσ N )} ˆ σD ) tr{N t r(N = p ˆ M Mˆ p :

λˆ = from which it is obtained that

: : ˆσ ˆσ )ˆ tr(N ) , D p = t r(N ˆ λ= p p N M M

(12.64)

where

ˆ σ N) M p = Mˆ p − tr (N

(12.65)

The substitution of Eq. (12.64) into Eq. (12.62), one has :

σD = σ + tr(Npσ ) σ N ˆ M

:

(12.66)

Then, the strain rate is given by D = E −1 σD + D p = E −1{σ + :

:

:

ˆ σ) ˆ σ) tr(N t r(N ˆ N p σ N} + p M M

i.e. ˆ: ˆ D = E−1 σ: + tr(Npσ ) (N + E −1σN ) M

(12.67)

The positive proportionality factor Λˆ in terms of strain rate is given by Eq. (6.97) as it is. Then, noting σD = EDe , the Jaumann rate of Cauchy stress is given from Eq. (12.62) by :

σ = E(D − D p ) − Λˆ σ N

12.3 Plastic Spin

323

= ED − E

ˆ ED) ˆ ED) tr(N tr(N ˆ− σ N ˆ ˆ) ˆ ˆ) N Mˆ p + tr( NEN Mˆ p + tr( NEN

i.e.

σ = [E − ( EN + σ N )⊗ (NE) ]D ˆ ˆ p tr( ˆ ˆ ˆ) M + NEN

:

(12.68)

The stress and the internal variables are updated by

σ• = σ + Wσ −σW :

(12.69)

ˆ p )α −α ( W − W ˆ p ) ⎫⎪ α• = a || D p|| + ( W − W

⎬ ˆ p )β − β ( W − W ˆ p ) ⎪⎭ β= b || D p' || + ( W − W

(12.70)

•

noting Eq. (12.60). Hereinafter, limit to the Mises yield condition with the kinematic hardening. Then, substituting Eq. (6.56) (6.85) into Eq. (6.89) into Eq. (12.61), the plastic spin reduces to the following equation given by Dafalias (1985). ˆ p = ρ λˆ (σN ˆ −N ˆ σ) W 2

=

ρ ˆ ρ λ (σ σˆ ' − σˆ ' σ) = λˆ{(σˆ + α ) σˆ ' − σˆ ' (σˆ + α )}

=

ρ ˆ ρ ρ α σˆ ' − σˆ ' α) = 2 λˆ (αNˆ − Nˆ α) = 2 (αD p − D p α) 2λ(

|| σˆ ' ||

2

|| σˆ ' ||

|| σˆ ' ||

|| σˆ ' ||

2

|| σˆ ' ||

|| σˆ ' ||

(12.71) Then, considering the simple shear deformation and assuming the rigid plasticity, the initial isotropy and the linear kinematic hardening as in 12.2, we have • • ˆ p = ρ ⎡α11 α12 ⎤ ⎡0 1⎤ γ − ⎡0 1⎤ γ ⎡α11 α12 ⎤ W 2 ⎢α − α11 ⎥ ⎢1 0 ⎥ 2 ⎢1 0 ⎥ 2 ⎢α − α11 ⎥ ⎣ ⎦ ⎣ 12 ⎣ 12 ⎦⎣ ⎦ ⎦

(

=

)

ρ ⎡ α12 α11 ⎤ ⎡α12 − α11 ⎤ γ• ρ ⎡ 0 α11 ⎤ γ• − 2 ⎢ −α11 α ⎥ ⎢α11 α ⎥ 2 = 2 ⎢−α ⎥ 12 ⎦ 12 ⎦ 11 0 ⎣ ⎣

(

)

⎣

Substituting Eqs. (2.144) and (12.72) into Eq. (12.70)1, we have

⎦

(12.72)

324

12 Corotational Rate Tensor

⎡• • ⎤ ⎞ • ⎛ • ⎢α 11 α 12 ⎥ = 2 h a ⎡ 0 1⎤ γ + ⎜ ⎡ 0 1 ⎤ γ − ρ α 11 ⎡ 0 1 ⎤ γ• ⎟ ⎡α11 α12 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ −1 0 ⎥ ⎢α − α ⎥ ⎢α• − α• ⎥ 3 ⎣1 0 ⎦ 2 ⎜ ⎣ −1 0 ⎦ 2 2 11 ⎦ ⎣ ⎦ ⎟ ⎣ 12 11 12 ⎣ ⎦ ⎝ ⎠ ⎡α11 α12 ⎤ ⎛ ⎡ 0 1 ⎤ γ• ρ ⎡ 0 1 ⎤ • ⎞⎟ ⎜ α − −⎢ 11 ⎥ ⎢ ⎥ ⎢ −1 0 ⎥ γ ⎟ ⎣α12 − α11 ⎦ ⎜ ⎣ −1 0 ⎦ 2 2 ⎣ ⎦ ⎝ ⎠

⎡ α12 −α11 ⎤ • ⎡0 1⎤ • ⎡ α12 − α11 ⎤ γ• ρ − α 11 ⎢ = 13 h a ⎢ ⎥ γ + ⎢ ⎥ ⎥γ ⎣1 0 ⎦ ⎣ −α11 − α12 ⎦ 2 2 ⎣−α11 −α12 ⎦ ⎡ −α12 α11 ⎤ γ• ρ ⎡−α12 α11 ⎤ • + α 11 ⎢ −⎢ ⎥ ⎥γ 2 2 ⎣ α11 α12 ⎦ ⎣ α11 α12 ⎦ ⎡α −α11 ⎤ • ⎡0 1⎤ • ⎡ α12 − α11 ⎤ • γ − ρα 11 ⎢ 12 = 13 h a ⎢ ⎥ γ + ⎢ ⎥ ⎥γ − − − α α α 1 0 12 ⎦ ⎣ ⎦ ⎣ − 11 ⎣ 11 α12 ⎦

⎡ 1 h a − (1 − ρ α11 )α11 ⎤ ⎢ (1 − ρ α 11)α12 ⎥• 3 =⎢ ⎥γ ⎢ 1 h a − (1 − ρ α11 )α − (1 − ρ α 11)α ⎥ 11 12 ⎣⎢ 3 ⎦⎥

(12.73)

from which we obtain d α11 = (1 − ρα 11)α 12 dγ

⎫ ⎪⎪ ⎬ d α12 = 1 h a − (1 − ρα11 )α ⎪ 11 3 dγ ⎪⎭

(12.74)

The nonlinear differential equation (12.74) is solved numerically by Dafalias (1985). The calculation result is shown in Fig. 12.5. As seen in this figure, the non-oscillation curve is obtained by choosing the material parameter ρ appropriately. When choosing ρ > 0.5 , σ 11(= −σ 22) and σ12 increase monotonically with the increase of shear strain γ . On the other hand, the Jaumann and the Green-Naghdi rates are independent of material property and thus they would lack the physical pertinence. The simple shear of the hypoelastic material and the kinematic hardening material exhibit the oscillatory shear stress if the Jaumann rate is adopted as described above. On the other hand, the simple shear of the isotropic elastoplastic material does not exhibit oscillatory shear stress for any corotational rate since the center of yield surface is fixed in the origin of stress space and the stress continues to lie on the yield surface in the plastic deformation process.

12.3 Plastic Spin

325

4

σ 12

ρ =1

3

3

σ 11

ρ = 0.5

ρ = 0.5

2

2

ρ = 0.42 ρ = 0.42

1

0

2

ρ =0

ρ = 0.3

ρ =0

0

ρ =1

1

6

4

8

10

γ

0

0

4

2

(a )

ρ = 0.3

6

8

γ

10

(b)

3 α 12 2

ρ =1 ρ = 0.5

1 ρ = 0.42

0

−1 0

ρ =0

ρ = 0.3

1

2

γ

3

(c)

Fig. 12.5 Description of simple shear deformation of kinematic hardening material by the corotational rate with a plastic spin (Dafalias, 1985)

Chapter 13

Localization of Deformation 13 Localization of Deformation

Even if material is subjected to a homogeneous stress, the deformation concentrates in a quite narrow strip zone as the deformation becomes large and finally the material results in failure. Such a concentration of deformation is called the localization of deformation and the strip zone is called the shear band. The shear band thickness is the order of several microns in metals and ten and several times of particle radius in soils. Therefore, the large shear deformation inside the shear band is hardly reflected in the change of external appearance of the whole body. Therefore, a special care is required for the interpretation of element test data and the analysis taking accounting of the inception of shear band is indispensable when a large deformation is induced. The localization phenomenon of deformation and its pertinent analysis are described in this chapter.

13.1 Element Test The element test of material is useful on the premise that a homogeneous deformation proceeds reflecting the constitutive property. However, the deformation becomes heterogeneous when a large deformation accompanying with σ 11

M p< 0

σ 11

Constitutive property

d σ 11

Element test

0 D11 dt 11

D11 dt 11

³ D11dt

Fig. 13.1 Stress-strain curve in the constitutive property and the element test with a localization in a softening state K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 327–336. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

328

13 Localization of Deformation

the shear band is induced. On the other hand, the state of stress (rate) is almost identical outside and inside the shear band, since the equilibrium equation is fulfilled. Therefore, the strain for a given stress is measured far smaller in the element test than that predicted in the constitutive relation reflecting the mechanical property. This fact is illustratively depicted in Fig. 13.1 for the softening material. This fact must be considered when we determine the material parameters from element test data.

13.2 Gradient Theory The material in which the stress at a certain material point is uniquely determined only by the history of the deformation gradient F at that point is called the simple material. However, when the gradient of deformation gradient F becomes large as seen inside the shear band, its history influences on the state of stress (rate). Here, there exist the limitation in the gradient so that the shear band thickness does not decrease less than a certain limitation. The limitation is regulated by the material constant, called the characteristic length. In the finite element method ignoring the influence of the gradient, the deformation concentrates in the narrow band zone corresponding to the width of one element. Then, if the finite elements are downsized aiming at obtaining an accurate solution, the shear band thickness reduces infinitely, resulting in the mesh-size dependence losing the reliability of solution, i.e. the il-posedness. The theory for the simple material ignoring the gradient is called the local theory. On the other hand, the theory taking account of the gradient is called the non-local theory. The non-local theory of elastoplastic deformation is first proposed by Aifantis (1984). Adopting the yield condition with the isotropic hardening for sake of simplicity and introducing the gradient of the isotropic hardening variable, let the yield condition (6.30) be extended as follows:

f (σ ) = 〈 F ( H ) 〉 where

〈〉

(13.1)

designates the second-order gradient, i.e. 2 2 〈 〉 =1 + c22 ∇ 2 , ∇ 2 ≡ ∂ 2 + ∂ 2 + ∂ 2 2

∂x1

∂x2

∂x3

(13.2)

c2 is the material constant describing the effect of the gradient of the mechanical state. Here, for the sake of simplicity, the higher order gradient is not incorporated. The first-order gradient is not incorporated because the odd-order gradients cancel each other in opposite directions. The material-time derivative of Eq. (13.1) is given by tr{

• ∂ f (σ ) D σ } = F' (1 + c22 ∇ 2) [H ] ∂σ

(13.3)

13.2 Gradient Theory

329

Substituting the associated flow rule (6.48) into Eq. (13.3), one has t r{

∂ f (σ ) D σ } = F' h( σ, Hi , N )(1 + c22 ∇ 2)[ λ ] ∂σ

(13.4)

from which we obtain

(1 − c22 ∇ 2)[σD ]) λ = tr(N p

(13.5)

M

tr(N(1 − c22 ∇ 2)[σD ]) N Mp { (1 − c22 ∇ 2)[σD ]} t r( NED) = t r( N σD ) + t r(NEN ) tr N Mp D

(13.6)

D = E−1 σ +

≅ {M p (1 + c22 ∇ 2) + t r(NEN)}[

2 D tr{N(1 − c2 ∇ 2)[σ]} ] Mp

2 D tr{N(1 − c2 ∇ 2)[σ]} Mp 2 [ ] } ∇ M p + t r(NEN) Mp D c2 (13.7) )[ tr{N(1 − 2∇)[σ ]}] = {M p + t r(NEN)}(1 + ∇ p M

= {M p + t r(NEN )}{1 + c22

where

≡ ϑ c22 ∇ 2 , ϑ ≡ ∇

Mp M p + t r( NEN )

(13.8)

On the other hand, it is obtained from Eq. (13.7) that

∇)D) ) [ M t +r(NED ] ≅ tMr(NE+ t(1r(−NEN t r(NEN) )

) Λ = (1 − ∇

p

(13.9)

p

)[D]) (1 − ∇ σD = ED − E t r(NE N p

(13.10)

M + t r(NEN)

In what follows, the above-mentioned equations are extended for the subloading surface model with the isotropic and the anisotropic hardening. Incorporating the gradient into the internal variables, Eq. (7.33) for the subloading surface is extended as (Hashiguchi and Tsutsumi, 2006):

f (σ − 〈α〉, 〈β〉 ) = 〈 RF ( H ) 〉

(13.11)

330

13 Localization of Deformation

Considering Eq. (13.2), Eq. (13.11) leads to f (σ − (1 + c2 ∇ 2)[α ] , (1 + c2 ∇ 2) [β]) = (1 + c2 ∇ 2) [ RF ( H )] 2

2

2

(13.12)

The material-time derivative of Eq. (13.12) is given by tr{

2 2 2 ∂ f (σ − (1 + c2 ∇ )[α ] , (1 + c2 ∇ 2)[β]) D σ} ∂σ

− tr{ ∂

+t r{

f (σ − (1 + c22 ∇ 2)[α ] , (1 + c22 ∇ 2)[β ]) (1 + c22 ∇ 2)[ αD ]} ∂σ

2 2 2 D ∂ f (σ − (1 + c2 ∇ 2)[α ] , (1 + c2 ∇ )[β ]) (1 + c22 ∇ 2) [β]} 2 2 c ∂ (1 + 2 ∇ ) [β ]

•

•

2 2 = (1 + c2 ∇ )[ R F + R F ]

(13.13)

The gradients of internal variables can be ignored since they are small compared to the gradient of their rates and thus Eq. (13.13) reduces approximately to tr{ ∂

D f (σˆ , β) D f (σˆ , β) (1 + c2 2)[ D f (σˆ , β) 2∇ σ } − tr{ ∂ α]} + tr{ ∂ β (1 + c22 ∇ 2) [β]} ∂σ ∂σ ∂ •

•

= F (1 + c22 ∇2) [R ] + R F' (1 + c22 ∇2) [H ]

(13.14)

Substituting Eq. (7.13) for the evolution rule of normal-yield ratio, the consistency condition is derived from Eq. (13.14) as follows: tr{ ∂

ˆ , β) D ˆβ f (σ f (σˆ , β) D] (1 + c22 ∇ 2)[α σ } − tr{ ∂ } + tr{ ∂ f (∂σβ, ) (1 + c22 ∇ 2) [βD ]} ∂σ ∂σ •

= UF (1 + c22 ∇ 2) [ D p ] + R F' (1 + c22 ∇ 2 ) [H ]

(13.15)

Further, substituting the associated flow rule (7.19) into Eq. (13.15), it is obtained that tr{ ∂

f (σˆ , β) D f (σˆ , β) f (σˆ , β) (1 + c22 ∇ 2)[λ b]} (1 + c22 ∇ 2)[λa]} + tr{ ∂ σ } − tr{ ∂ ∂σ ∂σ ∂β

= UF (1 + c22 ∇ 2) [ λ ] + R F' (1 + c22 ∇ 2) [λ h ]

(13.16)

13.3 Shear-Band Embedded Model: Smeared Crack Model

331

which can be approximately given by

{ ∂ f (∂σˆσ, β )σD } − tr{ ∂ f (∂σˆσ, β ) a(1 + c ∇ )[λ]}+ tr{ ∂f (∂σˆβ, β ) b(1 + c ∇ )[λ]} 2 2

tr

2 2

2

2

= (UF + RF' h)(1 + c22 ∇ 2)[ λ ]

(13.17)

The positive proportionality factor is derived from Eq. (13.17) as

λ = (1 − c22 ∇ 2)[

t r( N σD ) t r( N(1 − c22 ∇ 2)[σD ]) ≅ p M Mp

]

(13.18)

where ∂f (σˆ , β) ′ b))σ M p ≡ tr N a + ( F h + U − 1 tr ( RF ∂β F R

[{

}] .

(13.19)

On the derivation of Eq. (13.18), it is assumed that the fourth-order gradient can be ignored, thereby leading to (1 + c22 ∇ 2)(1 − c22 ∇ 2) ≅ 1 . Consequently, the plastic strain rate is given as Dp =

2 tr(N(1 − c2 ∇2)[ıD ]) N Mp

(13.20)

The shear band thickness of softening soil has been predicted adopting Eq. (13.20) by Hashiguchi and Tsutsumi (2006). Here, it is noteworthy that we must use quite small elements with the size of several tens of shear band thickness to take the effect of the gradient into account correctly in the finite element analysis. Therefore, it is nearly impossible to apply the gradient theory to the finite element analysis of boundary value problems in engineering practice at least at present. The gradient theory is used widely for prediction of shear band thickness, size effects, etc. using fine meshes for very small specimens.

13.3 Shear-Band Embedded Model: Smeared Crack Model Although the gradient theory is not applicable to the analysis of practical engineering problems at present, the practical model for the finite element analysis for softening materials has been proposed, as described below. As the deformation becomes large and the shear band is formed, the plastic deformation concentrates in the shear band and thus the softening is accelerated leading to the rapid reduction of stress. As the result, inversely, the unloading leading to the elastic state occurs inside the shear band. Consequently, the

332

13 Localization of Deformation

elastoplastic constitutive equation holds only inside the shear band. Then, denoting the strain rate and the plastic strain rate calculated from the external appearance by D and D p , respectively, called the apparent strain rate and the apparent plastic strain rate, respectively, the elastoplastic constitutive equation in terms of the apparent strain rate is proposed. It is called the shear-band embedded model or smeared crack model (Pietrueszczak and Mroz, 1981; Bazant and Cedolin, 1991). Denoting the ratio of the area of a shear band to the area of a finite element by S ( 1) , the following relations hold. p D p = SD

(13.21)

D+S D = De + D p = De + S D p = E−1σ

D

tr(N σ) N Mp

(13.22)

Tanaka and Kawamoto (1988) proposed the simple equation of S for the plane strain condition as follows:

S = ( w × l ) /(l × l ) = w / Fe

(13.23)

supposing simply the square finite element l×l with the side-length l and the shear band having the thickness w , where Fe ( = l × l ) is the area of the finite element. The positive proportionality factor is expressed in terms of the apparent strain rate from Eq. (13.22) as follows: tr(N ED) M p + tr( NEM ) S

Λ=

(13.24)

Then, the stress rate is given by EN σD = ED − M p + tr( NE N) S t r(N ED)

(13.25)

Then, we have D

tr(N σ) =

Mp tr(NED) + M S tr( NEN) p

(13.26)

from which it is known that the stress reduction is accelerated when the shear band is induced resulting in S < 1 . It is desirable to choose material parameters such that Eq. (13.22) or (13.25) fits to a measured stress-strain curve using the value of S predicted by a pertinent method, if we determine them from element test data.

13.4 Necessary Condition for Shear Band Inception

333

13.4 Necessary Condition for Shear Band Inception Discontinuity of the velocity gradient is induced at the shear band boundary. Here, incorporate the coordinate system in which the coordinate axes x1∗ and x2∗ are taken to be normal and parallel, respectively, to the shear band as shown in Fig. 13.2. The discontinuity of velocity gradient can be induced only in the x1∗ -direction. Therefore, only the following quantities are not zero, designating the difference by the discontinuity as Δ ( ) .

g11∗ ≡ Δ ( ∂v∗1 ) , g12∗ ≡ Δ ( ∂v∗2 ) ∂x1 ∂x1

(13.27)

Then, the discontinuity of strain rate is given by

v j ∂x∗ ∂vj v v ∂x∗ Δ Dij = 1 {Δ ( ∂ i ) + Δ ( ) } = 1 {Δ ( ∂ ∗i ) r + Δ ( ∂ ∗ ) r } 2 2 x x x x ∂ j ∂ i ∂ r ∂ j ∂xr ∂xi ∂vj ∂x∗ v ∂x∗ = 1 {Δ ( ∂ ∗i ) 1 + Δ ( ∗ ) 1 } = 1 {g1i∗ (n • e j ) + g1∗j (n • ei )} 2 2 ∂x1 ∂x j ∂x1 ∂xi = 1 ( g1i∗ n j + g1∗j ni ) 2

(13.28)

where n is the unit vector in the direction normal to the shear band, i.e. the x1∗ -direction. On the other hand, the discontinuity in the rate of traction vector t n applied to the discontinuity surface of velocity gradient, having the direction vector n , is described by

1 ⎫ • Δ t n1 = Δσ j1 n j = C ej1pkl ΔDkl n j = C ej1pkl 2 ( g1k∗ nl + g1l∗ n k ) n j ⎪ ⎪ (13.29) ⎬ 1 • Δ t n2 = Δσ j 2 n j = C ej 2pkl ΔDkl n j = C ej 2pkl 2 ( g1∗k nl + g1l∗ n k ) n j ⎪ ⎪⎭ Noting

1 1 C ej1pkl 2 ( g1k∗ nl + g1l∗ n k ) n j = 2 (C ej1pkl g1k∗ nl n j + C ej1pkl g1l∗ n k n j ) 1 p g1 p g2 n n + C ej12 n n + C ej1pklg11∗ n k n j + C ej1pk 2g1∗2 n k n j ) = 2 (C ej11 l 1∗ l j l 1∗ l j 1 p g1 p g2 p g1 n n + C ej12 n n + C ej11 n n + ep g 2 n n = 2 (C ej11 l 1∗ l j l 1∗ l j l 1∗ l j C j12l 1∗ l j ) p g1 p g2 n n + C ej12 nn = (C ej11 l 1∗ l j l 1∗ l j )

334

13 Localization of Deformation

Eq. (13.29) is expressed as ep ep ep ep 1 1 ⎧ • ⎫ ⎪Δ t n1 ⎪ ⎡C j11l nl n j C j1k 2 n k n j ⎤ ⎪⎧ g1∗ ⎪⎫ ⎡C j11l nl n j C j12 k n k n j ⎤ ⎪⎧ g1∗ ⎪⎫ ⎥ ⎨ ⎬ = ⎢ ep ⎥⎨ ⎬ ⎨ ⎬ = ⎢ ep ep ep 2 2 ⎪⎩Δ •t n2 ⎪⎭ ⎢⎣C j 21l nl n j C j 2k 2 n k n j ⎥⎦ ⎪⎩ g1∗ ⎪⎭ ⎢⎣C j 21l nl n j C j 22k n k n j ⎥⎦ ⎪⎩ g1∗ ⎪⎭

That is to say, 1 ⎧Δ •t ⎫ ⎪ n1 ⎪ ⎡ A11 A12 ⎤ ⎪⎧ g1∗ ⎪⎫ • • j = ⎨ ⎬ ⎢ ⎥ ⎨ 2 ⎬ (Δ tn i = Aij g1∗ , Δ t n = A g 1∗) • A A g ⎪⎩Δ t n2 ⎪⎭ ⎣ 21 22 ⎦ ⎪⎩ 1∗ ⎪⎭

(13.30)

where Aij is given by the following equation and called the acoustic tensor. ep A ≡ nCepn, Aij ≡ Crijs nr ns

(13.31)

Here, noting that the traction rate vector must be continuous, i.e. Δt n = 0 by the equilibrium and thus it must hold from Eq. (13.30) that 1 ⎡ A11 A12 ⎤ ⎧⎪ g1∗ ⎫⎪ ⎧0 ⎫ j ⎢ A A ⎥ ⎨ 2 ⎬ = ⎨0 ⎬ (Aij g1∗ = 0, A g1∗ = 0 ) ⎣ 21 22 ⎦ ⎪⎩ g1∗ ⎭⎪ ⎩ ⎭

(13.32)

In order that Eq. (13.32) has a solution other than the non-trivial solution g1∗ = 0 , i.e. that the discontinuity of velocity gradient is induced, it must hold that

det A = 0

(13.33)

The eigenvalue of the acoustic tensor A becomes zero as known from Eq. (1.127) when (13.33) holds. The search for the occurrence of n fulfilling Eq. (13.33), i.e. the inception of the shear band and its direction, is called the eigenvalue analysis. Equation (13.33) is given explicitly as (Hashiguchi and Protasov, 2004) p det(nCep n) = det(C1eijsp n1ns + C2eijs n2 ns )

= det(C1eijp1n1n1 + C1eijp2 n1n2 + C2eijp1n2 n1 + C2eijp2 n2 n2 ) =

=

ep ep ep ep ep ep ep ep C1111 n1n1 + C1112 n1n2 + C2111 n2 n1 + C2112 n2 n2 C1121 n1n1 + C1122 n1n2 + C2121 n2 n1 + C2122 n2 n2 ep ep ep ep ep ep ep ep C1211 n1n1 + C1212 n1n2 + C2211 n2 n1 + C2212 n2 n2 C1221 n1n1 + C1222 n1n2 + C2221 n2 n1 + C2222 n2 n2

ep 2 ep ep ep 2 ep 2 ep ep ep 2 C1111 n1 + (C1112 + C2111 )n1n2 + C2112 n2 C1121 n1 + (C1122 + C2121 )n1n2 + C2122 n2 ep 2 ep ep ep 2 ep 2 ep ep ep 2 C1211 n1 + (C1212 + C2211 n2 C1221 n1 + (C1222 + C2221 n2 )n1n2 + C2212 )n1n2 + C2222

13.4 Necessary Condition for Shear Band Inception

335

ep 2 ep ep ep 2 ep 2 ep ep ep 2 = {C1111 n1 + (C1112 + C2111 ) n1n2 + C2112 n2 }{C1221 n1 + (C1222 + C2221 ) n1n2 + C2222 n2 } ep 2 ep ep ep 2 ep 2 ep ep ep 2 − {C1121 n1 + (C1122 + C2121 ) n1n2 + C2122 n2 }{C1211 n1 + (C1212 + C2211 ) n1n2 + C2212 n2 }

ep ep ep ep = (C1111 C1221 − C1121 C1211 )n14 ep ep ep ep ep ep ep ep 　+ (C1111 C1222 + C1111 C2221 +C1112 C1221 + C2111 C1221 ep ep ep ep ep ep ep ep − C1121 C1212 − C1121 C2211 − C1122 C1211 − C2121 C1211 )n13n2 ep ep ep ep ep ep ep ep + (C1111 C2222 + C1221 C2112 − C1121 C1211 − C1211 C2122 )n12 n22 ep ep ep ep ep ep ep ep + (C1122 C2222 + C2111 C2222 + C1222 C2112 + C2221 C2112

ep ep ep ep ep ep ep ep − C1122 C2212 − C2121 C2212 − C2122 C1212 − C2122 C2211 )n1n23 ep ep ep ep + (C2112 C2222 − C2122 C2212 )n24 ep ep ep 2 4 = (C1111 C1212 − C1112 )n1 ep ep ep ep ep ep ep ep 　+ (C1111 C1222 + C1111 C2212 − C1112 C1122 − C1122 C1211 )n13 n2 ep ep ep ep ep ep ep ep + (C1111 C2222 + C1212 C1212 − C1112 C1211 − C1211 C1222 )n12 n22 ep ep ep ep ep ep ep ep + (C1122 C2222 + C12111 C2222 − C1122 C1222 − C1222 C1122 )n1n23 ep ep ep 2 + (C1212 C2222 − C1222 )n24

which reduces to

det A = a1n14 + a2 n13 n2 + a3 n12 n22 + a4 n1 n23 + a5 n24 = 0

(13.34)

where

⎫ ⎪ ep ep ep ep ep ep ep ep a2 ≡ C1111 C1222 + C1111 C2221 − C1121 C2211 − C1122 C1211 , ⎪ ⎪ ep ep ep ep ep ep ep ep a 3 ≡ C1111 C2222 + C1221 C2112 − C1121 C1211 − C1211 C2122, ⎬ (13.35) ⎪ ep ep ep ep ep ep ep ep a 4 ≡ C1122 + C2111 − C1122 C2222 C2222 C2212 − C2122 C2211 , ⎪ ep ep ep 2 ⎪ a 5 ≡ C2112 C2222 − C2122 ⎭ ep ep ep 2 a1 ≡ C1111 C1221 − C1121 ,

　

Setting

n1 = cosθ , n 2 = sinθ

(13.36)

336

13 Localization of Deformation

x2∗

x1∗

v

x2

n 0∗

e2

0

e1

x1

Fig. 13.2 Discontinuity of velocity gradient induced in the direction normal to the shear band

Eq. (13.34) is rewritten as

g (θ ) = a5 tan 4 θ + a4 tan 3 θ + a3 tan 2 θ + a2 tan θ + a0 = 0 (13.37) which, noting the symmetry g (θ ) = g (−θ ) , leads to g (θ ) = a5 tan 4 θ + a3 tan 2 θ + a0 = 0

(13.38)

There exists the possibility that a shear band occurs in the direction θ fulfilling Eq. (13.38). Here, note that Eq. (13.33) is the only necessary condition for the inception of the shear band. We searched above the discontinuity of the velocity gradient in the direction normal to the shear band, while the traction rate vector must be continuous in that direction. Inversely, on the other hand, the search for the discontinuity of the normal stress rate component applied to the surface normal to the shear band, i.e. Δ σ• ∗22 ≠ 0 , while the normal strain rate component in the direction parallel to the shear band must be zero, i.e. Δ ε• ∗22 = 0 , is called the compliance method (cf. Mandel, 1964). Here, note that the normal strain rate component in the direction normal to the shear band can be ∗ discontinuous, i.e. Δ ε• 11 ≠ 0 in dilatable materials. It was applied to the prediction of the direction of shear band formation in soils (Vermeer, 1982).

Chapter 14

Numerical Calculation Numerical Calculatio n

Elastoplastic deformation has to be analyzed generally by numerical calculations with finite incremental steps since the constitutive equation is given in the rate (incremental) form. In particular, the return-mapping algorithm to pull back the stress to the yield surface must be incorporated into the computer program adopting the conventional elastoplastic constitutive models. On the other hand, the subloading surface model is furnished with the distinguished advantage for the numerical calculation with the automatic controlling function to attract the stress to the normal-yield surface in the plastic deformation process and thus it does not require to incorporate the convergence computer algorithm such as the return mapping in the normal-yield state. Nevertheless, it requires to incorporate the return-mapping algorithm in the subyield state since the subyield state is out of the stress-controlling function. Basic equations for the return mapping method extended to the subloading surface model is described in this chapter.

14.1 Numerical Ability of Subloading Surface Model The stress controlling function of the subloading surface model is described in Chapter 7. This fact will be qualitatively shown below in the numerical calculation by the concise examination for the response of the uniaxial loading behavior, adopting the simplest version, i.e. the initial subloading surface model for the isotropic Mises material with the evolution rule of the normal-yield ratio • R = u cot{(π /2) R}||D p || in Eq. (7.13) with Eq. (7.15) with Re = 0 . The response of the conventional elastoplastic constitutive model is also shown for the comparison. The relations of the axial stress σ a and the normal-yield ratio R versus the axial strain ε a are depicted in Fig. 14.1. The responses for the linear isotropic hardening F = F0 + hcε ep ( hc : material constant) are depicted in Fig. 14.1(a) and those for the nonlinear isotropic hardening in Eq. (10.3) are shown in Fig. 14.1(b). The two levels of axial strain increment dε a = 0.0006 and 0.0055 are input for numerical

calculations. Here, any special stress controlling algorithm to pull it back to the yield surface is not introduced. The material parameters are chosen as follows: K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 337–347. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

338

14 Numerical Calculation

1000 0.0055

d ε a =0.0006

800

600 σa (MPa)

0.0055

d ε a =0.0006

400

200

0 2.0

1.50

dε a =0.0006

d ε a = 0.0055

1.0

d ε a = 0.0055 d ε a =0.0006

R 0.50

0

0.02

0.04

εa

0.06

0.08

(a) Linear isotropic hardening Exact curve of conventional elastoplasticity elastoplasticit Calculated by the conventional elastoplasty elastoplast Calculated by the subloading surface mod Fig. 14.1 Numerical accuracies of the conventional elastoplastic and the subloading surface model: Uniaxial loading behavior of Mises material with isotropic hardening.

14.1 Numerical Ability of Subloading Surface Model

339

1000

0.0055 d ε a = 0.0006

800

600 σa (MPa)

0.0055 d ε a = 0.0006

400

200

0

2.0

1.50 0.0055

0.0006

1.0 0.0055

R

d ε a = 0.0006 0.50

0

0.02

0.04

εa

0.06

(b) Nonlinear isotropic hardening

Fig 14.1b. (continued)

0.08

340

14 Numerical Calculation

Material constants:

Youg's modulus: E = 100000MPa, ⎧Linar isotropic : h c = 7000, Hardening ⎨ ⎩ Nonlinar isotropic : h1 = 0.8, h 2 = 50, Evolution of normal - yield ratio : u = 200. Initial values:

Hardening function: F0 = 500MPa, Stress : σ 0 = 0MPa The nonsmooth curves bent at the yield stress are expressed by the conventional model. Moreover, the stress deviates from the exact curve of concventional elastoplasticity. The deviation becomes large with the increases in the nonlinearity of hardening and in the input strain increment. On the other hand, the stress is automatically attracted to the normal-yield surface in the subloading subloading surface model even for the quite large strain increment dε a = 0.0055 (0.55%) . The zigzag lines tracing the exact curve are calculated such that the stress rises up when the it lies below the normal-yield surface but it drops down immediately if it goes over the normal-yield surface, obeying the evolution rule of normal-yield ratio in • • Eq. (7.13) with Eq. (7.14), i.e. R > 0 for R<0 and R < 0 for R >0 . The amplitude of zigzag decreases gradulally in the monotonic loading process, while, needless to say, the amplitude is smaller for a smaller input increment of strain. Eventually, the subloading surface model posseses the distinguished high ability for numerical calculation as verified also quantitatively in these concrete examples, which has not been attained in any other elastoplastic constitutive equations incuding the multi, the two, the infinite, nonlinear-kinematic hardening and the bounding surface models delineated in Chapters 7 and 8.

14.2 Return-Mapping Formulation for Subloading Surface Model Elastoplastic constitutive equation is described by simultaneous equations of stress rate components and strain rate components which do not fulfill the complete integrability condition and thus elastoplastic deformation analysis is performed by numerical calculations with finite steps. Therefore, an error is accumulated in the calculation using the Euler method (Yamada et al., 1968) without the convergence calculation if we adopt the conventional elastoplastic constitutive models in which the interior of yield surface is assumed to be the purely-elastic domain. Then, various return-mapping algorithms, e.g. the mean normal method (Rice and Tracey,

14.2 Return-Mapping Formulation for Subloading Surface Model

341

1973; Pillinger et al., 1986) and the radial return method (Krieg and Krieg, 1977) with the correction algorithm so as to pull the stress back to the yield surface, have been proposed to date. However, they are intended for the conventional elastoplasticity premised on the assumption that the interior of yield surface is an elastic domain. The subloading surface model is furnished with the controlling function to attract the stress to the normal-yield surface automatically as described in 14.1 and thus it has no need to incorporate the return-mapping algorithm to pull back the stress to the normal-yield surface. However, the subyield state before the stress reaches the normal-yield surface is out of this controlling function and thus the calculation error is accumulated in cyclic loading process in which the subyield state ( R < 1) is repeated. An improvement in numerical calculation so as to reflect exactly the evolution of subloading surface would be required in the subyield state. The return-mapping formulation for the subloading surface model will be given in this section, while the Jaumann rate is adopted for the corotational rate for sake of simplicity. As known from Eq. (7.26), the stress increment in elastoplastic constitutive equation is given by subtracting the plastic relaxation stress increment due to plastic strain rate from the elastic stress increment calculated presuming that the input strain increment is elastic. Here, the stress rate is related to the plastic strain rate by the consistency condition (7.18) for the subloading surface (7.6). Suppose that the stress σ n in the step n is calculated already by giving n-time inputs of the strain increment. Then, for the initiation of calculation in the n + 1 step, first calculate the elastic stress increment dσ elastic using the elastic constitutive n+1 equation for D n +1 dt and Wn +1 dt due to the input velocity gradient increment Ln+1 dt , i.e.

dσ elastic n +1 = (E (σ n ) Dn +1 + Wn +1σ n − σ n Wn +1 ) dt

(14.1)

noting Eq. (4.41). Here, dσ elastic is called the elastic predictor which is predicted n+1 as a trial supposing the elastic deformation process. Then, the stress is updated by adding it to the stress σ n in the last step, i.e. elastic σelastic n+1 = σ n + dσ n+1

(14.2)

It can be stated that the calculation as the elastic deformation was correct if the elastic is smaller than the subloading surface f (σ elastic n+1 − α n , β n ) passing through σ n +1 is subloading surface f (σ n − α n , βn ) at the end of step n . Then, σ elastic n +1 determined as the stress at the n + 1 step. On the other hand, if f (σ elastic − α n , β n ) is n+1 larger than f (σn − α n , βn ) , it is regarded that a plastic strain rate is necessarily induced during the input strain increment Dn +1 dt , i.e.

342

14 Numerical Calculation Final elastic ⎫ p f ( σ elastic n +1 − α n , β n ) ≤ R n F ( H n ) or ReF ( H n ) : Dn+1 = 0, σ n +1 = σ n +1 ⎪

Otherwise

⎬

elastic p (1) : Dn+1 ≠ 0, σ Final n+1 ≠ σ n +1 ⎪ ⎭

(14.3) In the second case of Eq. (14.3) the plastic relaxation stress increment must be (k) (k) (k) (k) (k) . Presuming that σ n+ reduced from σ elastic n +1 1 , H n +1 , α n+1 , β n+1 , Rn+1 are already obtained repeating calculations of stress reduction by k time calculations, then let the k + 1-th calculation be carried out. Designating the plastic strain increment p (k+1)δ t in this time, the plastic relaxation stress increment p (k+1) , called the Dn+1 δ σ n+1 plastic corrector, is given by p (k+1) p (k+1) δ t (k+1) σ (k) = E(σ (k) n+1 ) D n+1 n+1 − σ n +1 = dσ n+1

(14.4)

Here, it is desired that the following relation is fulfilled. (k+1) (k+1) (k+1) ˆ (k+1) f (σ n+1 , β n+1 ) − Rn+1 F ( H n+1 ) = 0

(14.5)

(k+1) Applying the Taylor expansion to H n(k+1+1) , α(kn+1+1) , H(kn +1+1) and Rn+1 in Eq. (14.5), one has

ˆ n(k)+1 , β n(k)+1 ) + t r{∂ f (σ

(k) ) (k) ) ˆ (k) ˆ (k) f (σ f (σ (k) n +1 , β n +1 n +1 , β n +1 δ σ p n+1} − t r{∂ δ α (k+1) (k) (k) n+1 } ∂ σ n +1 ∂ σ n +1

+ t r{ ∂

(k) ) ˆ (k) f (σ n +1 , β n + 1 } δ β (k+1) n+1 ∂ β (k) n +1

(k+1) { (k) + (k) ) + F ( H (k) ) H (k+1) } ) F ( H n+1 − ( Rn+1 δ Rn+1 ' n+1 δ n+1

∂ (k) ˆ (k) ≅ f (σ n +1 , β n +1 ) + t r{

+ t r{∂

(k) ) ˆ (k) ˆ (k) , β (k) ) f (σ f (σ n +1 , β n +1 δ p (k) σ n+1} − t r{∂ n +1(k) n +1 δ α (k+1) (k) n+1 } ∂ σ n +1 ∂ σ n +1

(k) ) ˆ (k) f (σ n +1 , β n +1 δ β (k) n+1} (k) ∂ β n +1

(k+1) (k+1) F ( H (k) ) − R (k) F ( H (k) ) − Rn(k)+1 F ( H n(k)+1) − δ Rn+1 n +1 n +1 ' n +1 δ H n +1 = 0

(14.6)

where from Eqs. (6.37) and (6.86) one has (k) (k) ; p (k+1) δ H n(k+1) +1 = h (σ n+1 , Hi n+1 D n +1 δ t )

(14.7)

14.2 Return-Mapping Formulation for Subloading Surface Model

343

(k) p (k+1) (k) δ α (k+1) n +1 = a(σ n+1 , Hi n+1)|| D n +1 ||δ t

(14.8)

(k) p (k+1) (k) δ β(k+1) n +1 = b(σ n+1 , Hi n+1)|| D 'n +1 ||δ t

(14.9)

(k +1) δ Rn(k+1+1) = U ( Rn(k)+1 ) || D pn(k+1+1) ||δ t , U ( Rn(k)+1 ) = u cot ( π 〈R n +1 − R e 〉) 2 1 − Re (14.10) Substituting Eqs. (14.7)-(14.10) into Eq. (14.6), the consistency condition is obtained as follows: (k) ) ˆ (k) ∂ f (σ (k) p (k +1) n +1 , β n +1 ˆ (k) f (σ δ t} E(σ (k) n+1 ) D n+1 n +1 , β n +1 ) + t r{ (k) ∂ σ n +1

−t r{∂

(k) ) ˆ (k) f (σ n +1 , α n +1 k) , H (k) )|| p (k+1) || t } a(σ (n+1 δ i n+1 D n +1 (k) ∂ σ n +1

+ t r{ ∂

(k) ) ˆ (k) f (σ n +1 , β n +1 k) , H (k) )|| p (k+1) || δ t} b(σ (n+1 i n+1 D 'n +1 (k) ∂ β n +1

1) (k) − Rn(k)+1F ( H n(k)+1) − U ( Rn(k)+1 ) || D pn(k+ +1 δ t || F ( H n +1)

p (k+1) k) , H (k) ; − Rn(k)+1F' ( H n(k)+1) h (σ (n+1 i n+1 D n +1 δ t ) = 0

(14.11)

The associated flow rule is expressed from Eq. (7.35) as

ˆ (k+1) ˆ ˆ (k) (k) D pn(k+1) +1 = λ n+1 N ( σ n+1 , β n+1 )

(14.12)

Substituting Eqs. (14.12) into Eq. (14.11), it holds that

∂ (k) ˆ (k) f (σ n +1 , β n +1 ) + t r{

− t r{∂

(k) ) ˆ (k) f (σ n +1 , β n +1 (k+1) (k) ˆ ˆ (k) E(σ (k) n+1 )λ n +1 δ t N ( σ n+1 , β n +1 )} ∂ σ (k) n +1

(k) ) ˆ (k) f (σ k) n +1 , α n +1 k) , a(σ (n+1 )λ (k+1) Hi (n+1 n+1 δ t )} ∂ σ (k) n +1

(k) ) ˆ (k) f (σ n +1 , β n +1 k) , H (k) ) (k +1) ||N || t ) b (σ (n+1 i n +1 λ n +1 'δ } (k) β ∂ n +1 (k) − Rn(k)+1F ( H n(k)+1) − U ( Rn(k)+1 ) λ (k+1) n+1 δ t F ( H n +1)

+ t r{ ∂

(k) k) , H (k) ; λ (k+1) ˆ (σ ˆ (k) − Rn(k)+1F' ( H n(k)+1)h (σ (n+1 n+1 , β n+1 ) ) = 0 i n+1 n+1 δ t N

(14.13)

344

14 Numerical Calculation

from which one has (k) (k) (k) (k+1) ˆ (k) λ n+1 δ t = { f (σ n +1 , β n +1 ) − Rn+1F ( H n+1) }

ˆ ,β ∂ f (σ /[− t r{ ∂ σ

(k) (k) n +1 n +1 ) (k) n +1

k) ) ˆ (σ (k) , H (k) ) N n+1 i n+1 } E(σ (n+1

(k) ) ˆ (k) f (σ (k) n +1 , β n +1 a(σ (k) , + t r{∂ n+1 Hi n+1)} (k) ∂ σ n +1

− t r{∂

(k) ) ˆ (k) f (σ n +1 , β n +1 k) , H (k) ) ||N || b (σ (n+1 i n+1 '} ∂ β(k) n +1

(k) (k) (k) (k) k) ; ˆ (σ (k) k) , ) h (σ (n+1 Hi (n+1 + Rn+1 F' ( H n+1 N ˆ (k) n+1 , β n+1 ) ) + U ( R n+1) F ( H n+1)

]

(14.14) By use of (k) ) (k) f (σˆ (k) ∂ f (σˆ (k) n+1 , β n+1 ) (k) n +1 , β n +1 ˆ (σˆ (k) N = n+1 , β n+1 ) (k) (k) ˆ (k) ˆ (σˆ (k) ∂ σ n +1 t r {N n+1 , β n+1 )σ n+1}

(k) n+1 (k) n+1

(14.15)

(k) n+1) (k) ˆ (σˆ (k) N n+1 , β n+1 ) (k) ) ˆ (k) n+1 σ n+1}

(≠ t r {NˆR(σˆ F (, βH

)

based on Eq. (7.20), Eq. (14.14) is expressed as (k) (k) (k) (k+1) ˆ (k) λ n+1 δ t = { f (σ n +1 , β n +1 ) − Rn+1 F ( H n+1)} (k) ˆ (k) f (σ (k) (k) ˆ (k) n+1 , β n+1 ) (k) ˆ (σ ˆ (k) t r {N n+1 , β n+1 )E(σ n+1) N (σ n+1 , Hi n+1)} (k) (k) ) ˆ (k) β , σ } n+1 n+1 n+1

/ [− t r {Nˆ (σˆ +

(k) ˆ (k) f (σ n+1 , β n+1 ) (k) (k) (k) ˆ (σ ˆ (k) t r{N n+1 , β n+1 ) a(σ n+1 , Hi n+1)} (k) ˆ (k) ˆ (σ ˆ (k) t r {N n+1 , β n+1 )σ n+1}

(k) ) ˆ (k) f (σ n +1 , β n +1 k) , H (k) )||N || − t r{∂ b (σ (n+1 i n+1 '} (k) ∂ β n +1

(k) (k) (k) (k) (k) (k) k) , H (k) ; ˆ (σ + Rn+1 F' ( H n+1 ) h (σ (n+1 i n+1 N ˆ n+1 , β n+1 ) ) + U ( R n+1) F ( H n+1)

]

14.2 Return-Mapping Formulation for Subloading Surface Model

345

Then, one has (k) (k) (k) ˆ ˆ (k) (k) ˆ (k) ˆ (k) F ( H n+1 {1 − Rn+1 ) / f (σ (k+1) n+1 , β n+1 )} t r{N (σ n+1 , β n+1 )σ n+1} λ n+1 δt= p (k) (k) (k) (k) (k) (k) ˆ ˆ ˆ σ ˆ n+1 , β n+1 )E(σ n+1) N( n+1 , β n+1) } + Mn+1 −t r {N(σ

(14.16) where (k) (k) (k) (k) ˆ (σ ˆ (k) M pn+1 = t r{N n+1 , β n+1 ) a(σ n+1 , Hi n+1)}

[

(k) (k) k) ; ˆ (σ (k) k) , Hi (n+1 + Rn+1F' ( H n+1)h (σ (n+1 N ˆ (k) n+1 , β n+1 ))

(k) ) ˆ (k) f (σ (k) (k) (k) n +1 , β n +1 b (σ (k) , − t r{∂ n+1 Hi n+1)||N' ||}+ U ( R n+1) F ( H n+1) (k) β ∂ n +1 (k) ˆ (k) ˆ (σ ˆ (k) t r {N n+1 , β n+1 )σ n+1} (k) ˆ n+1 , β (k) f (σ ) n+1

] (14.17)

Then, the plastic strain rate is obtained from Eqs. (14.12) and (14.16) as follows: 1) D pn(k+ +1 δ t =

(k) F ( H (k) ) (k) ˆ (k) (k) ˆ ˆ (k) ˆ (k) {1− Rn+1 n+1 / f (σ n+1 , β n+1 )σ n+1 , β n+1 )} t r{ N(σ n+1} p (k) (k) (k) (k) (k) (k) ˆ ˆ ˆ ˆ n+1, βn+1 )E(σ n+1)N( σ n+1, βn+1) } + Mn+1 − t r {N(σ (k) ˆ (σ ˆ (k) N n+1 , β n+1 )

Here, σ, H , α, β and

εp

(14.18)

are updated from Eqs. (14.4) and (14.7)-(14.9) by

p (k+1) (k) (k) (k) (k) σ (k+1) n+1 = σ n+1 − E(σ n+1 ) D n +1 δ t + ( Wn +1 σ n+1 − σ n+1Wn +1 )δ t (14.19)

(k) p (k+1) (k) (k) H n(k+1) +1 = H n +1 + h (σ n+1 , Hi n+1; D n +1 δ t )

(14.20)

(k) (k) (k) (k) p (k+1) (k) α (k+1) n +1 = α n +1 + a (σ n+1 , Hi n+1)|| D n +1 ||δ t + ( Wn +1α n +1 − α n +1Wn +1 )δ t

(14.21) (k) (k) (k) p (k+1) (k) β (k) β (k+1) n +1 = β n +1 + b (σ n+1 , Hi n+1) || D n +1 ||δ t + ( Wn +1 β n +1 − n +1Wn +1 )δ t

(14.22)

ε np+1(k +1) = ε np+1(k)+ || D pn(k+1+1) ||δ t

(14.23)

where

ε p ≡ ∫ || D p ||δ t 1) is given explicitly by Eq. (14.18). D pn(k+ +1

(14.24)

346

14 Numerical Calculation

(k+1) has to be calculated from Eq. (7.16) as Rn+1

ε p −ε p − (k+1) R n+1 = π2 (1 − Re )cos −1{cos(π Rn Re )exp (− π u n+1 n )} + Re

2 1 − Re

2

1 − Re

(14.25) (k+1) On the other hand, R n+1 must be calculated by the numerical integration of Eq.

(14.10), if we adopt an evolution rule of the normal-yield ratio in which the analytical relation of R to ε p does not hold. The calculation must be continued until the error of the following equation becomes less than a certain limit as shown in Fig. 14.2. p p − −1 π ε n+1 − ε n (k+1) 2 ˆ (k+1) cos π Rn Re f (σ n +1 , β n+1 ) − π (1 − Re )cos { ( 2 1 − Re ) exp (− 2 u 1 − Re )}

[

]

(k+1) +Re F ( H n+1 )=0

(14.26)

which is obtained by substituting Eq. (14.25) into (14.5).

Stress elastic ı n+ 1 p (1) δ ın+ 1 p (2) δ ı n+ 1 p (3) δ ı n+ 1

elastic dı n+ 1

ˆ , ȕ) = RF ( H ) f (ı

ın

0

Dn +1 dt

Strain

Fig. 14.2 Return-mapping process attracting the stress to the subloading surface

Basic equations of the return-mapping method are shown above for the subloading surface model which is the only pertinent unconventional model

14.2 Return-Mapping Formulation for Subloading Surface Model

347

fulfilling the continuity and smoothness conditions. On the other hand, the bounding surface model with a radial-mapping of Dafalias, which has somewhat similar structures to the subloading surface model, is incapable of incorporating the return-mapping method since no loading surface passing through the current stress point is assumed and thus its evolution rule is not incorporated in the subyield state. The return mapping method executed so as to fulfill the equation of the subloading surface is shown above for the initial subloading surface model. It can be formulated for the extended subloading surface model in the similar way. The stress is attracted to the normal-yield surface and thus an accurate calculation would be executed in normal-yield state by adopting small incremental steps even if the return-mapping algorithm is not adopted.

Chapter 15

Constitutive Equation for Friction 15 Constitutive Equatio n for Frictio n

All bodies in the natural world are exposed to friction phenomena, contacting with other bodies, except for bodies floating in a vacuum. Therefore, it is indispensable to analyze friction phenomena rigorously in addition to the deformation behavior of bodies themselves in analyses of boundary value problems. The friction phenomenon can be formulated as a constitutive relation in a similar form to that of the elastoplastic constitutive equation of materials. A constitutive equation for friction with the transition from the static to the kinetic friction and vice versa and the orthotropic and rotational anisotropy is described in this chapter.

15.1 History of Constitutive Equation for Friction Formulation of the friction phenomenon as a constitutive equation was first attained for a rigid-plasticity (Seguchi et al., 1974; Fredriksson, 1976). Subsequently, it was extended to an elastoplasticity (Michalowski and Mroz, 1978; Oden and Pires, 1983a,b; Curnier, 1984; Cheng and Kikuchi, 1985; Oden and Martines, 1986; Kikuchi and Oden, 1988; Wriggers et al., 1990; Wriggers, 2003; Peric and Owen, 1992; Anand, 1993; Mroz and Stupkiewicz, 1998; Gearing et al., 2001) in which the elastic springs between contact surfaces is incorporated. In them the isotropic hardening is introduced to describe the test results (cf. Courtney-Pratt and Eisner, 1957) exhibiting the smooth contact traction vs. sliding displacement curve reaching static-friction. However, the interior of the sliding-yield surface has been assumed as an elastic domain. Therefore, the plastic sliding velocity induced by the rate of traction inside the sliding-yield surface is not described. Needless to say, the accumulation of plastic sliding displacement induced by the cyclic loading of contact traction within the sliding-yield surface cannot be described by these models. They might be called the conventional friction model in accordance with the classification of plastic constitutive models by Drucker (1989). On the other hand, based on the concept of the subloading surface, the subloading-friction model (Hashiguchi et al., 2005b; Ozaki et al, 2007) falling within the framework of unconventional plasticity, called the unconventional friction model, was formulated, which describes the smooth transition from the elastic to plastic sliding state and the accumulation of sliding displacement during a cyclic loading of tangential contact K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 349–385. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

350

15 Constitutive Equation for Friction

traction. Additionally, in this model, the reduction of friction coefficient with the increase of normal contact traction observed in experiments (cf. e.g. Bay and Wanheim, 1976; Dunkin and Kim, 1996; Gearing et al., 2001) is described by incorporating the nonlinear sliding-yield surface. Besides, the decrease has not been taken into account for Coulomb sliding-yield surface, which has been adopted widely to date as a constitutive models for friction. It is widely known that when bodies at rest begin to slide against one other, a high friction coefficient appears first, which is called the static friction. Subsequently, a friction coefficient decreases approaching a stationary value, called the kinetic friction. Furthermore, if the sliding ceases for a while and then starts again, the friction coefficient recovers and similar behavior to that of the initial sliding is reproduced (Dokos, 1946; Rabinowicz, 1951, 1958; Howe et al., 1955; Derjaguin, et al., 1957; Brockley and Davis, 1968; Kato et al., 1972; Richardson and Noll, 1976; Horowitz and Ruina, 1989; Ferrero and Barrau, 1997; Bureau et al., 2001). The recovery of friction coefficient has been formulated using equations including the time elapsed after the stop of sliding (cf. Rabinowicz, 1951; Howe et al., 1955; Brockley and Davis, 1968; Kato et al., 1972; Horowitz and Ruina, 1989; Bureau et al., 2001). However, the inclusion of time itself leads to the loss of objectivity in constitutive equations, since the evaluation of elapsed time after the stop of sliding is accompanied with arbitrariness, as known from the state in which a sliding velocity varies at a slow level. On the other hand, generally speaking, the variation of material property has to be described in terms of the sliding velocity, the stress and internal variables. The reduction of the friction coefficient from the static to kinetic friction and the recovery of the friction coefficient as described above are the fundamental characteristics in friction phenomena, which have been widely recognized for a long time. Difference of the static and kinetic frictions often reaches up to several tens of percent, and thus the formulation taken account of these characteristics is of importance for analyses of practical problems in engineering. In addition, difference of friction coefficients is observed in mutually opposite sliding directions. That difference can be described by the rotation of a sliding-yield surface, whereas the anisotropy of soils has been described by the rotation of a yield surface, as described in 11.4. Further, the difference of the range of friction coefficients is observed in different sliding directions. It would be describable by the concept of orthotropy of the sliding-yield surface (Mroz and Stupkiewicz, 1994).

15.2 Decomposition of Sliding Velocity The sliding velocity v is defined as the relative velocity of the counter body and is expressed by the normal part vn and the tangential part v t as follows (see Fig. 15.1):

v = v n + vt

(15.1)

15.2 Decomposition of Sliding Velocity

351

f Counter body

fn

n

vt v

ft

vn

Fig. 15.1 Contact traction f and sliding velocity

v

where

v n = ( v • n )n = (n ⊗ n ) v = −vn n , ⎫ ⎬ vt = v − v n = (I − n ⊗ n ) v. ⎭

(15.2)

n is the unit outward-normal vector at the contact surface. vn is the normal component of the sliding velocity, i.e.

vn ≡−n • v where the sign of

(15.3)

vn is selected to be plus when the counter body approaches the

relevant body. Here, it is assumed that v is additively decomposed into elastic sliding velocity e v and plastic sliding velocity v p , i.e. v = ve + v p

(15.4)

Then, vn and v t are expressed by the elastic and the plastic parts as follows: p vn = v en + v n , ⎫⎪ ⎬ p vt = v et + v t , ⎪⎭

(15.5)

where ven = ( ve • n)n = (n ⊗ n) ve = −vnen,⎫⎪ ⎬ vte = ve − ven = (I − n ⊗ n) ve, ⎪⎭

(15.6)

352

15 Constitutive Equation for Friction

v np = ( v p • n)n = (n ⊗ n) v p = −vnpn,⎪⎫ ⎬ p vt = v p − v np = (I − n ⊗ n) v p. ⎪⎭

(15.7)

p vne and vn are the elastic and the plastic part, respectively, of vn in Eq. (15.3). The contact traction f acting on the body is expressed by the normal traction f n

and the tangential traction f t as follows:

f = fn + ft ,

(15.8)

where f n ≡ (n • f )n = (n ⊗ n)f = − fn n ,⎫⎪ ⎬ f t ≡ f − fn = (I − n ⊗ n)f = ft t ⎪⎭

(15.9)

whilst n and t are the unit vectors in the directions of f n and f t , i.e.

n≡

fn f , ≡ t || f n || t || f t ||

(15.10)

and fn and ft are the magnitudes of f n and f t , i.e.

fn ≡ −|| f n || = −n • f , ft ≡ || f t ||

(15.11)

while the sign of fn is selected to be plus when the relevant body is compressed by the counter body. Here, note that the directions of the tangential contact traction and the tangential sliding velocity are not necessary identical in general and thus I − n ⊗ n ≠ t ⊗ t in the three-dimensional space. Here, it holds that

⎫ ∂fn ∂{(n ⊗ n) • f } ∂f n ∂ (− n • f ) = = n, = = − n ,⎪ ∂f ∂f ∂f ∂f ⎪ ⎬ ∂f t ∂{(I − n ⊗ n) • f } ⎪ n n − I ⊗ = = ∂f ∂f ⎪⎭

(15.12)

fn (I − n ⊗ n) − f t ⊗ (− n) I − n ⊗ n + η ⊗ n (15.13) ∂η ∂ ft / fn = = = fn f n2 ∂f ∂f ∂||η|| ∂ ||η|| = ∂f ∂η

•

∂η I − n ⊗ n + η⊗ n 1 = {( τ • η ) n + τ } = τ• fn fn ∂f (15.14)

where η ≡ ft fn

(15.15)

15.2 Decomposition of Sliding Velocity

vdt t + dt

B' vB dt

353

A'

f (t + dt )

vA dt

D

f (t )

f dt ȍfdt

t

•

f dt

A B f (t )

Fig. 15.2 Objectivities of the sliding velocity

τ≡

D

v and the corotational traction rate f

η (n • τ = 0 ) ||η||

(15.16)

Now, let the elastic sliding velocity be given by the following hypo-elastic relation, whilst the elastic sliding velocity is usually far smaller than the plastic sliding velocity in the friction phenomenon. D 1 D v en = α f n , v et = 1 f t α n

(15.17)

t

where fD n and fD t are the normal component and tangential component, respecD tively, of f , which are related to the material-time derivative as follows: D

•

D

D

•

•

f = f − Ωf , f n = f n − Ωfn , f t = f t − Ωf t

(15.18)

which is derived from D

•

•

•

D

D

f = f − Ωf = (fn + f t )• − Ω(f n + f t ) = f n − Ωfn + f t − Ωf t = f n + ft (15.19) where the skew-symmetric tensor Ω is the spin describing the rigid-body rotation of the contact surface. α n and α t are the contact elastic moduli in the normal and the tangential directions to the contact surface. On the other hand, the sliding velocity v is not an absolute velocity of a point on the body surface but the relative

354

15 Constitutive Equation for Friction

velocity between two points on the contact surface, independent of the rigid-body rotation of contact surface (see Fig. 15.2), and thus it can be adopted to the constitutive relation having the objectivity. It follows from Eq. (15.17) that D

D

D

f = f n + f t = Ce v

e

(15.20)

where the second-order tensor C e is the contact elastic modulus tensor given by Ce = α n n ⊗ n + αt ( I − n⊗ n), ⎫⎪ ⎬ −1 1 1 Ce = α n ⊗ n + α ( I − n ⊗ n ) ⎪ n t ⎭

(15.21)

15.3 Normal Sliding-Yield and Sliding-Subloading Surfaces Assume the following sliding-yield surface with the isotropic hardening/softening and the rotational isotropy, which describes the sliding-yield condition.

f (f , β ) = F

(15.22)

where F is the isotropic hardening/softening function denoting the variation of the size of the sliding-yield surface. The anisotropy of the friction phenomenon would be substantially described by the rotation of yield surface similarly to the deformation of soils as a frictional material described in 11.4. β is the vector describing the rotation of the sliding-yield surface around the origin of traction space and it does not have the normal component since it describes the anisotropy. Hereinafter, D let β be assumed to be a fixed vector leading to β = 0 for sake of simplicity. Then, it holds that

n• β = 0 •

β= Ωβ .

(15.23) (15.24)

Here, for simplicity it is assumed that f (f , β ) is the homogeneous function of f in degree-one. Then, it holds that f (s f , β ) = s f (f , β )

(15.25)

∂f (f , β) • f = f (f , β ) ∂f

(15.26)

where s is an arbitrary positive scalar quantity, whilst Eq. (15.26) is based on Euler’s homogeneous function of degree-one. Therefore, the sliding-yield surface

15.4 Evolution Rules of Sliding-Hardening Function

355

retains a similar shape and orientation with respect to the origin of contact traction space, i.e. f = 0 for β = const. In what follows, we assume that the interior of the sliding-yield surface is not a purely elastic domain but that the plastic sliding velocity is induced by the rate of traction inside that surface. Therefore, let the sliding-yield surface be renamed as the normal sliding-yield surface. Then, based on the concept of the subloading surface (Hashiguchi, 1980, 1989), we introduce the sliding-subloading surface, which always passes through the current contact traction point f and retains a similar shape and orientation to the normal sliding-yield surface with respect to the origin of contact traction space, i.e. f = 0 . Let the ratio of the size of the sliding-subloading surface to that of the normal sliding-yield surface be called the normal sliding-yield ratio, denoted by R (0 ≤ R ≤ 1) , where R = 0 corresponds to the null traction state ( f = 0 ) as the most elastic state, 0 < R < 1 to the subsliding state ( 0 < f < F ), and R = 1 to the normal sliding-yield state in which the contact traction lies on the normal sliding-yield surface ( f = F ). Therefore, the normal sliding-yield ratio R plays the role of three-dimensional measure of the degree of approaching the normal sliding-yield state. Then, the sliding-subloading surface is described by

f (f , β) = RF .

(15.27)

The material-time derivative of Eq. (15.27) leads to D

•

•

N• f = R F + R F

(15.28)

where N≡

∂f (f , β ) ∂f

(15.29)

The direct transformation of the material-time derivative to the corotational derivative is verified by substituting Eq. (15.18) into Eq.(15.28), noting a • (Ωa ) = 0 for an arbitrary vector a . The general proof is given in Chapter 4. 15.4 Evolutio n Rules of Sliding-Harde ning F unctio n

15.4 Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio Evolution rules of the isotropic hardening function and the normal sliding-yield ratio are formulated so as to reflect experimental facts.

15.4.1 Evolution Rule of Sliding-Hardening Function The following might be stated from the results of experiments.

356

15 Constitutive Equation for Friction

1) The friction coefficient first reaches the maximal value of static-friction and then decreases to the minimal stationary value of kinetic-friction if the sliding commences. Physically, this phenomenon might be interpreted to result from separations of the adhesions of surface asperities between contact bodies because of the sliding (cf. Bowden and Tabor, 1958). Then, let it be assumed that the reduction results from the contraction of the normal sliding-yield surface, i.e., plastic softening caused by the sliding. 2) The friction coefficient recovers gradually with the elapse of time and the identical behavior as the initial sliding behavior exhibiting the static friction is reproduced if sufficient time elapses after the sliding ceases. Physically, this phenomenon might be interpreted to result from the reconstructions of the adhesions of surface asperities during the elapsed time under a quite high contact pressure between edges of surface asperities. Then, let it be assumed that the recovery results from the viscoplastic hardening because of the creep phenomenon. Taking account of these facts, let the evolution rule of the isotropic hardening/softening function F be postulated as follows (Hashiguchi and Ozaki, 2008): • n m F = −κ ( F − 1) || v p || + ξ (1 − F ) Fs Fk

(15.30)

where Fs and Fk (Fs ≥ F ≥ Fk) are the maximum and minimum values of F for the static and kinetic frictions, respectively. Both κ and m are the material constants influencing the decreasing rate of F because of the plastic sliding, and both ξ and n are the material constants influencing the recovering rate of F because of the elapse of time, whereas ξ is a function of absolute temperature in general. The first and the second terms in Eq. (15.30) stand for the deteriorations and the reformations, respectively, of the adhesions between surface asperities. On the other hand, these phenomena have been described by the softening because of the sliding displacement and the hardening due to the time elapsed after the stop of sliding up to the present. However, the inclusion of the time itself in constitutive equations is not allowed, as described in 15.1.

15.4.2 Evolution Rule of Normal Sliding-Yield Ratio It is observed in experiments that the tangential traction increases first elastically with plastic sliding and thereafter it gradually increases approaching the normal sliding-yield surface similarly as in the plastic deformation described in 7.2. Then, similarly to the evolution rule (7.13) of the normal-yield ratio R for the plastic deformation, we assume the evolution rule of the normal sliding-yield ratio as follows:

15.5 Relations of Contact Traction Rate and Sliding Velocity

357

•

R || v p ||

= U (R )

0

1

R

　

Fig. 15.3 Function U ( R ) for the evolution rule of the normal sliding- yield ratio R •

R = U ( R ) || v p || for v p ≠ 0

(15.31)

where U (R) is a monotonically decreasing function of R fulfilling the following conditions (Fig. 15.3).

U ( R ) → +∞ for R = 0, ⎫ ⎪ for R = 1, ⎬ U (R) = 0 (U ( R ) < 0 for R > 1). ⎪⎭

(15.32)

Let the function U satisfying Eq. (15.32) be simply given by U ( R ) = u cot(π2 R )

(15.33)

where u is the material constant. Equation (15.31) with Eq. (15.33) can be led to the analytical integration of R for accumulated plastic sliding u p ≡ ∫|| v p||dt under the initial condition u p = u 0p : R = R 0 as follows: R = π2 cos −1[cos(π R0 )exp{− π u (u p − u 0p )}]

2

(15.34)

2

15.5 Relations of Contact Traction Rate and Sliding Velocity The substitution of Eqs. (15.30) and (15.31) into Eq. (15.28) gives rise to the consistency condition for the sliding-subloading surface:

358

15 Constitutive Equation for Friction D

N•f = R

{ −κ ( FFk − 1) m || v p || +ξ (1 − FFs )n}+ U || v p || F

(15.35)

Assume that the direction of plastic sliding velocity is tangential to the contact plane and outward-normal to the curve generated by the intersection of the sliding-yield surface and the constant normal traction plane f n = const. , leading to the tangential associated flow rule v p = λ t n (λ > 0, || t n || = 1)

(15.36)

where λ ( > 0) is a positive proportionality factor and t n ≡ (I − n ⊗ n ) • N || ( I − n ⊗ n ) • N ||

(15.37)

Substituting Eq. (15.36) into Eq. (15.35), the proportionality factor λ  is derived as follows: D c (15.38) λ = N • f −p m m and thus D

• − mc vp = N f p tn m

(15.39)

where m

m p ≡ −κ ( F − 1) R + F U Fk

(15.40)

n m c ≡ ξ (1 − F ) R (≥ 0) Fs

(15.41)

Substituting Eqs. (15.20) and (15.39) into Eq. (15.4), the sliding velocity is given by

D

−1 D • f − mc v = Ce f + N m tn p

(15.42)

The positive proportionality factor in terms of the sliding velocity, denoted by the symbol Λ , is given from Eqs. (15.42) as • e• − m c Λ = Np C v e

(15.43)

m + N• C • tn

The traction rate is derived from Eqs. (15.4), (15.20), (15.36) and (15.43) as follows: D

( 〈mN +CN vC− mt 〉 t )

f = Ce v −

•

p

e

•

•

c e • n

n

(15.44)

15.6 Loading Criterion

359

15.6 Loading Criterion While the loading criterion for the plastic sliding velocity is similar to that for the plastic strain rate described in 6.3, it will be described below. First, note the following facts: 1. It is necessary that

λ =Λ>0

(15.45)

in the loading (plastic sliding) process v p ≠ 0 . 2. It holds that D

N•f ≤ 0

(15.46)

in the unloading (elastic sliding) process v p = 0 . Further, because of v = v e D

leading to N • Ce• v = N • Ce• v e = N • f in this process it holds that

Λ=

D

N• f − m c m p + N • Ce• t n

(15.47)

while it should be noted that m c ≥ 0 (Eq. (15.41)). 3. The plastic modulus m p takes both signs of positive and negative, while the first and the second terms in Eq. (15.40) are negative and positive, respectively. On the other hand, noting that the contact elastic modulus Ce is the positive definite tensor and thus it holds that N • C e• N m p in general and postulating that the plastic relaxation does not proceed infinitely, let the following inequality be assumed.

m p + N • Ce• t n > 0

(15.48)

Then, in the unloading process v p = 0 , the following inequalities hold depending on the sign of the plastic modulus m p , i.e. the hardening, perfectly-plastic and softening states from Eqs. (15.38) and (15.46)-(15.48).

λ ≤ 0 and Λ ≤ 0 when m p > 0

⎫ ⎪ λ → −∞ or indeterminate and Λ ≤ 0 when m p = 0 ⎬ ⎪ λ ≥ 0 and Λ ≤ 0 when m p < 0 ⎭

(15.49)

Therefore, the sign of λ at the moment of unloading from the state m p ≤ 0 is not necessarily negative. On the other hand, Λ is negative in the unloading process.

360

15 Constitutive Equation for Friction

Consequently, the distinction between a loading and an unloading processes cannot be judged by the sign of λ  but can be done by that of Λ . Therefore, the loading criterion is given as follows:

v p ≠ 0 : Λ > 0,

⎫⎪ ⎬ v p = 0 : otherwise. ⎪⎭

(15.50)

v p ≠ 0 : N • Ce • v − m c > 0, ⎫⎪ ⎬ v p = 0 : otherwise. ⎪⎭

(15.51)

or

on account of Eq. (15.48).

15.7 Sliding-Yield Surfaces It can be stated from experiments that the friction coefficient decreases with the increase of contact pressure (cf. Bay and Wanheim, 1976; Dunkin and Kim, 1996; Gearing et al., 2001; Stupkiewicz and Mroz, 2003). Therefore, the normal sliding-yield surface cannot be described appropriately by the Coulomb sliding-yield surface in which the tangential contact traction is linearly related to the normal contact traction through the constant friction angle. In what follows, the sliding-yield surface with the nonlinear relation of tangential contact traction and normal contact traction is assumed, by which the reduction of friction coefficient with the increase of normal contact traction is described. The closed normal sliding-yield and the sliding-subloading surfaces can be described by putting f (f , β ) = fn g ( χˆ )

(15.52)

as follows:

fn g ( χˆ ) = F , fn g ( χˆ ) = RF

(15.53)

where

χˆ ≡

||ηˆ || ˆ , η ≡ η −β M

(15.54)

M is the material constant denoting the traction ratio η ( = ft / fn ) at the maximum

point of ft . Simple examples of the function g ( χ ) in the sliding-yield function in Eq. (15.52) are as follows: g ( χˆ ) = exp( χˆ ), g ' ( χ ) = exp( χˆ )

(15.55)

15.7 Sliding-Yield Surfaces

361

g ( χˆ ) = 1 + χˆ 2 , g ' ( χˆ ) = 2 χˆ

(15.56)

g ( χˆ ) = exp( χˆ 2 /2), g ' ( χ ) = χˆ exp( χˆ )

(15.57)

g ( χˆ ) =

1 , g ( χˆ ) 1 = ' 1 − χˆ /2 2(1 − χˆ /2)2

(15.58)

All sets of Eqs. (15.22) and (15.52) with Eqs. (15.55)-(15.58) exhibit closed surfaces passing through points fn = 0 and fn = F at ft = 0 for β = 0 . Equation (15.55) and (15.56) are based on the original Cam-clay yield surface (Schofield and Wroth, 1968) and the modified Cam-clay yield surfaces (Roscoe and Burland, 1968), respectively, for soils described in 11.2. Equation (15.57) exhibits a teardrop-shaped surface (Hashiguchi, 1972, 1985; Hashiguchi et al., 2005b) which is reversed from the surface of Eq. (15.55) on the axis of normal contact traction. Equation (15.58) exhibits a parabola (Hashiguchi et al., 2005b). The normal sliding-yield and the sliding-subloading surfaces are depicted in Fig. 15.4 for Eq. (15.57) having the teardrop shape. Normal sliding- yield surface

ft

Sliding-subloading surface M 1

f 0

RF/ e

F/ e

RF

F

fn

Fig. 15.4 Teardrop shaped normal sliding-yield and sliding-subloading surfaces

It holds for Eqs. (15.13) and (15.54) 2 that ∂ηˆ ∂η I − n ⊗ n + η ⊗ n = = fn ∂f ∂f

∂||ηˆ || ∂ ||ηˆ || = ∂f ∂ ηˆ

•

(15.59)

∂ ηˆ I − n ⊗ n + η⊗n 1 = f {(τˆ • η ) n + τˆ } (15.60) = τˆ • fn ∂f n

362

15 Constitutive Equation for Friction

where

τˆ ≡

ηˆ (n • τˆ = 0) ||ηˆ ||

(15.61)

Further, it holds from Eqs. (15.12), (15.21), (15.52), (15.54), (15.59) and (15.60) that

∂χˆ ∂ ||ηˆ || / M = ∂f ∂f

= 1 {(τˆ • η ) n + τˆ } M fn

(15.62)

N = −g ( χˆ ) n + fn g' ( χˆ ) 1 {(τˆ • η ) n + τˆ } M fn g ( χˆ ) g χˆ = −{g ( χˆ ) − ' ( τˆ • η )}n + ' ( ) τˆ M M

(15.63)

(∂∂fft =) (I − n ⊗ n) • N g χˆ g ˆ g ˆ = (I − n ⊗ n ) • − {g ( χˆ ) − ' ( ) ( τˆ • η )}n + ' ( χ ) τˆ = ' ( χ ) τˆ M M M (15.64)

[

]

t n = τˆ g ( χˆ ) g χˆ ( τˆ • η )}n + ' ( ) τˆ N • C e = − {g ( χˆ ) − ' M M

[

(15.65)

]

•{α n n ⊗ n + α t ( I − n ⊗ n )}

g ( χˆ ) g ˆ ( τˆ • η )}n + α t ' ( χ ) τˆ = −α n{g ( χˆ ) − ' M M g' ( χˆ ) g ˆ ( τˆ • η )}n + α t ' ( χ ) τˆ N • Ce • t n = −α n{g ( χˆ ) − M M

[

(15.66)

] • τˆ = αt g'M( χˆ ) (15.67)

The substitution of Eqs. (15.21) and (15.62)-(15.67) into Eqs. (15.42) and (15.44) leads to the sliding velocity vs. contact traction rate and its inverse relation are given as follows: D 1 1 v ={α n n ⊗ n + α t ( I − n ⊗ n )} f

15.7 Sliding-Yield Surfaces

363

g ˆ −{g ( χˆ ) − ' ( χ ) ( τˆ η )}n + g' ( χˆ ) τˆ] fD − m [ M M + •

•

mp

c

τˆ

D = ( α1 − α1 )(n • f ) n + α1 fD n t t

g ˆ g ˆ −{g ( χˆ ) − ' ( χ ) ( τˆ • η)}(n • fD ) + ' ( χ ) ( τˆ • fD ) − m c M M τˆ + mp

(15.68)

D

f ={α n n ⊗ n + α t ( I − n ⊗ n )} g' ( χˆ ) α g χˆ

[

v−

〈

[− n{ (

)−

g ( χˆ ) ( τˆ • η )}n + α t ' τˆ M M g ( χˆ ) m p + αt ' M

]• v − m c

〉 τˆ]

= (α n − α t )(n • v) n + αt v g χˆ g ˆ − α n{g ( χˆ ) − ' ( ) ( τˆ • η )}(n • v) + α t ' ( χ ) ( τˆ • v) − m c M M α − t τˆ ˆ) p α g' ( χ m + t M

〈

〉

(15.69) On the other hand, the normal sliding-yield and the sliding-subloading surfaces for the circular cone of the Coulomb friction condition are given by putting

f (f , β ) = ||ηˆ ||, F =μ

(15.70)

||ηˆ || = μ , ||ηˆ || = R μ

(15.71)

as follows:

where μ is the friction coefficient and the evolution rule is given in an identical form with Eq. (15.30) as follows: • μ μ = −κ{( μμ ) m − 1}|| v p || + ξ {1 − (μ

k

s

)n }

(15.72)

μs and μ k signify material constants for the maximum and the minimum friction, i.e. static and kinetic friction coefficients, respectively. f (f , β ) in Eq. (15.70) is the homogeneous function of f in degree-zero. The normal sliding-yield

364

15 Constitutive Equation for Friction

ft

No

rm

a

r su ld e i g -y d in i l ls in S l id

fa

ce

g-su

b loa

s urf d in g

a ce

ff

0

fn

Fig. 15.5 Normal sliding-yield and sliding-subloading surfaces for Coulomb friction condition

and sliding-subloading surfaces in Eq. (15.71) are open surfaces having a conical shape, whereas they expand/contact with the increase/decrease of μ and R as shown in Fig. 15.5. It holds for Eq. (15.71) that μ

m p ≡ −κ{( μ

) m −1}R+ μU

(15.73)

k

μ

m c ≡ ξ{1 − ( μ

s

)n }R (≥ 0)

(15.74)

in stead of Eqs. (15.40) and (15.41). Further, it holds from Eqs. (15.12), (15.21), (15.60) and (15.70) that ||ηˆ || = 1 {(τˆ • η ) n + τˆ } N= ∂ fn ∂f

(15.75)

15.8 Basic Mechanical Behavior of Subloading-Friction Model

(∂∂fft =) (I − n ⊗ n) • N = (I − n ⊗ n) • N • Ce =

365

1 {(τˆ • η ) n + τˆ } = 1 τˆ (15.76) fn fn

1 {(τˆ • η ) n + τˆ } • {α n ⊗ n + α ( − n ⊗ n )} 1 {α (τˆ • η ) n +α τˆ } = n t t I fn fn n (15.77)

}• τˆ = αfnt

N • Ce • t n = 1 α n (τˆ • η) n +α t τˆ fn

{

(15.78)

for Coulomb friction condition. The substitution of Eqs. (15.21) and (15.75)-(15.78) into Eqs. (15.42) and (15.44) leads to the relation of sliding velocity vs. contact traction rate as follows:

1 {(τˆ • η ) n + τˆ } D m c •f − 1 n ⊗ n + 1 ( I − n ⊗ n ) D fn v ={α n }f + τˆ αt p m D 1 {(τˆ • η ) (n • f ) + ( τˆ • fD )} − m c D D fn 1 1 1 = (α − α )(n • f ) n + α f + τˆ n mp t t

(15.79)

The inverse relation of Eq. (15.79) is given as follows:

D

( 〈

f ={α n n ⊗ n + α t ( I − n ⊗ n )} v −

= (α n − αt )(n • v) n + αt v −

αt fn

1 {α (τˆ • η ) n +α τˆ }• v − m c t fn n m p + αt fn

〈α (τˆ n

• η)(n • v)

〉 τˆ)

+ αt ( τˆ • v) − m c

m p + αt fn

〉 τˆ (15.80)

15.8 Basic Mechanical Behavior of Subloading-Friction Model We examine below the basic response of the present friction model by numerical experiments and comparison with test data for the linear sliding phenomenon without a rigid-body rotation under a constant normal traction and with the fixed direction of tangential contact traction on the assumption of isotropy for sake of simplicity. Then, it holds that

366

15 Constitutive Equation for Friction

fn =const., t =const., Ω = 0, ⎫ ⎪⎪ • D D D • f n = 0, f t = ft t , f = ft t , ⎬ p v e = v tet , v p = v t t , v = v t t . ⎪ ⎪⎭

(15.81)

from Eqs. (15.4), (15.20) and (15.36).

15.8.1 Relation of Tangential Contact Traction Rate and Sliding Velocity Substituting Eqs. (15.81) into Eqs. (15.68) and (15.69), the tangential sliding velocity vs. contact traction rate and its inverse relation are given as follows:

vt =

{

1 α + t

g' ( χ ) / M −κ ( F − 1) R + U Fk m

f t = α t vt −

ft −

n ξ (1 − F ) R

Fs

M

(15.82)

−κ ( F − 1) R + U F Fk m

α t g' ( χ ) vt − ξ (1 − F F

{ 〈

•

} F

•

s

)nR

〉} g' ( χ )

(15.83)

−κ ( F − 1) R + U F + α t Fk M m

On the other hand, substituting Eqs. (15.81) into Eqs. (15.79) and (15.80), these relations for Coulomb friction condition are given as follows:

vt =

{α

1 + t

μ

1/ fn

}

m

−κ ( μ − 1) R + μ U k

•

μ n 1v fn t − ξ (1 − μ s ) R

{ 〈 −κ ( μ −1)

f t = α t vt −

μ n ξ (1 − μ s ) R • (15.84) ft − m μ −κ ( μ − 1) R + μ U k

μk

m

αt

R+ μ U + f n

〉}

(15.85)

The relation of the tangential contact traction ft vs. the tangential sliding displacement ut ( ≡ ∫v t d t ) is schematically shown in Fig. 15.6 for a high sliding velocity process in which the creep hardening of the second term in Eq. (15.82) is negligible. The relation by the conventional friction model with the sliding-yield surface enclosing an elastic domain is also shown as bold curves. In

15.8 Basic Mechanical Behavior of Subloading-Friction Model

367

the subloading-friction model, the softening term −κ ( F/Fk − 1) m R ( ≤ 0) increases monotonically from the negative value to zero and inversely the normal sliding-yield term U F ( ≥ 0) decreases monotonically from the infinite value to zero in the denominator of the plastic sliding velocity in second term in the bracket in Eq. (15.82). In the initial stage of sliding, the plastic modulus is positive, i.e. m p > 0 so that the tangential contact traction increases but thereafter these terms cancel mutually leading to m p = 0 at which the tangential contact traction reaches the peak, i.e. the static friction point p . Thereafter, the softening term decreases gradually to zero but the normal sliding-yield term decreases rapidly resulting in m p < 0 so that the tangential contact traction decreases to the kinetic friction point k . Initial normal sliding-yield surface at static friction

ft

ft

Final normal sliding-yield and subloading-sliding surfaces at kinetic friction state

y p

y p k

0

k ut

0

o Fk f n = const. Initial subloading-sliding surface

F0

fn

elastic ½ Conventional friction model elastoplastic ¾¿ Subloading-friction model

•

vp =

f m t , m p = − R κ ( F − 1) + U F mp Fk

p

o •

+

f

m − Rκ ( F − 1) + U F Fk

+∞

0

−

0

0

−

0

+

λ F

+

k

F0

(Softening)

Fk

Fig. 15.6 Prediction of linear sliding behavior from the static to the kinetic friction by the conventional friction and the subloading friction models at a high sliding rate without the creep hardening

15.8.2 Numerical Experiments and Comparisons with Test Data Numerical experiments and comparisons with test data for the subloading-friction model are shown below, whilst Eq. (15.83) with the teardrop shaped normal sliding-yield surface in Eq. (15.57) is adopted.

368

15 Constitutive Equation for Friction

0.8

vt = 1.0−5 mm/s

1.0−4

_

f ts / f n

0.6

_f tk / f n

ft / f n 0.4 0.2

1.0−1 1.0−2 1.0−3

0.0 0.0

0.1

0.2

ut (mm)

0

1.0 0.8

1.0−5 −4 1.00

0.6

−3 1.00

F / Fs 0.4 0.2 0.0 0.0

0−2 1.0 vt = 1.0−1 mm/s

0

0.1

0.2 0 200 400 600 Stationary time (s)

ut (mm)

Fig 15.7 Variations of the friction coefficient and hardening function with tangential sliding distance and stationary time for various tangential sliding velocity ( u =100mm-1)

For numerical experiments, the material parameters and the normal contact traction are selected as follows:

F0 = Fs = 100MPa, Fk = 30MPa, M = 0.3,

κ = 1 000MPa/mm, m = 1, ξ = 10MPa/s, n = 2, u = 5, 10, 50, 100 and 1 000mm −1 , α n = α t = 100MPa/mm, f n = 10MPa where F0 is the initial value of F . The evolution rule U ( R) = − u ln R of the normal sliding-yield ratio is used in stead of Eq. (15.33).

15.8 Basic Mechanical Behavior of Subloading-Friction Model

0.8

vt = 0.1mm/s

u = 1000mm−1

369

_

f ts / f n

0.6

_f tk / f n

ft / f n 0.4 100 50 10 5

0.2 0.0 0.0

0.8

0.05

0.10 0.15 ut (mm)

vt = 0.001mm/s

0.20

0.25

_

u = 1000mm −1

f ts / f n

0.6

ft / f n

_f tk / f n

0.4 100 50 10 5

0.2 0.0 0.0

0.8

0.05

0.10 0.15 ut (mm)

vt = 0.00001mm/s

0.20

0.25

_

s u = 1000mm −1 f t / fn

0.6

ft / f n

_f tk / f n

0.4 100 50 10 5

0.2 0.0 0.0

0.05

0.10 0.15 ut (mm)

0.20

0.25

Fig. 15.8 Influences of the material constant u in the evolution rule of normal sliding-yield surface on the relation of friction coefficient vs. tangential sliding displacement for three levels of tangential sliding velocity

370

15 Constitutive Equation for Friction 0.8

_

f ts / f n

0.6

ξ = 10

ft / f n

ξ =0

0.4

_f tk / f n

0.2

Stationary time: 1s 0.0 0.0

0.1

0.2 0.3 ut (mm)

0.4

0.5

0.8 _

f ts / f n

0.6

ξ = 10

ft / f n 0.4

ξ =0

0.2

Stationary time: 200s

_f tk / f n

0.0 0.0

0.1

0.2 0.3 ut (mm)

0.4

0.5

Fig. 15.9 Influence of stationary time on recovery of friction coefficient ( 　　　　　 ) ut = 0.01mm/s

Variations of the friction coefficient ft / f n with the tangential sliding displacement ut for various tangential sliding velocities are shown in Fig. 15.7, where ft s and f t k are the values of the tangential contract traction ft calculated from the normal sliding-yield surface for F = Fs and F = Fk , respectively, and the prescribed constant value of f n . The reduction of friction coefficient becomes remarkable with the increase of tangential sliding velocity vt . The reduction of the hardening function F with the tangential sliding displacement and its recovery with the time elapsed after the stop of sliding are also shown in this figure, while almost complete recover is realized for the stationary time of 400s. In addition, the influence of the material constant u in the evolution rule of normal sliding-yield ratio R on the friction coefficient is shown in Fig. 15.8. The

15.8 Basic Mechanical Behavior of Subloading-Friction Model

371

curve of tangential contract traction vs. tangential sliding displacement becomes smoother for the smaller vale of u inducing a larger plastic sliding velocity by the rate of contract traction inside the normal sliding-yield surface, whilst the conventional friction theory premising that the interior of sliding yield surface is an elastic domain is realized for u → ∞ . The influence of the stationary time on the recovery of the friction coefficient is shown in Fig. 15.9. The recovery is larger for a longer stationary time. The accumulation of sliding displacement under the cyclic loading of tangential contact traction of 80% of f t s is shown in Fig. 15.10, whilst it cannot be predicted by the conventional friction model ( u → ∞ ). It is shown that the accumulation proceeds more quickly as the sliding velocity increases.

0.8 0.6 ft / f n

_

vt = 0.00001mm/s

f ts / fn

Conventional model

80%

0.4 Present model

0.2 0.0 0.0 0.8

0.6

0.1

0.2 0.3 ut (mm)

vt = 0.1mm/s

0.4

0.5

_

f ts / f n

Conventional model

80%

ft / f n Present model

0.4 0.2

0.0 0.0

0.1

0.2 0.3 ut (mm)

0.4

0.5

Fig. 15.10 Accumulation of sliding displacement under cyclic loading for two levels of sliding velocity ( u =100mm-1)

372

15 Constitutive Equation for Friction 0.6 0.5 0.4

ft / f n

0.3 0.2 Experiment Present theory

0.1 0.0 0.0

0.02

0.04 0.06 ut (mm)

0.08

0.1

Fig. 15.11 Reduction process of friction coefficient from the static to kinetic-friction under the infinitesimal sliding velocity (Test data after Ferrero and Barrau, 1997)

The comparison with test data for the reduction process of friction coefficient from the static- to kinetic-friction is shown in Fig. 15.11. The test curve for sliding between roughly polished steel surfaces (Ferrero and Barrau, 1997) under the infinitesimal sliding velocity vt ≤ 0.0002 mm/s is simulated well enough by the present model, where the material parameters are selected as follows:

F0 = Fs = 120MPa, Fk = 25MPa, M = 0.28,

κ = 3 000MPa/mm, m = 2, ξ = 0.1MPa/s, n = 2, u = 1 500mm −1 ,

α n = α t = 10GPa/mm, f n = 10MPa, vt = 0.0002mm/s The comparison with test data for the recovery of friction coefficient by the stop of sliding on the way of the reduction process from the static- to kinetic-friction is depicted in Fig. 15.12. The test curves for sliding between roughly polished steel surfaces (Ferrero and Barrau, 1997) under the infinitesimal sliding velocity vt ≤ 0.0002 mm/s and the stationary time 20s and 400s are simulated sufficiently well by the present model, where the material parameters are selected as follows:

15.8 Basic Mechanical Behavior of Subloading-Friction Model

373

F0 = Fs = 120MPa, Fk = 30MPa, M = 0.28,

κ = 3 000MPa/mm, m = 2, ξ = 0.1MPa/s, n = 2, u = 1 500mm −1 ,

α n = α t = 100GPa/mm, f n = 10MPa, vt = 0.0002mm/s 0.6 Experiment Present theory (u = 3000 mm −1 )

0.5

ft / f n 0.4 Stationary time: 20s

0.3

0.2 0.0

0.01

0.02 0.03 ut (mm)

0.04

0.05

0.6 Experiment Present theory (u = 1500 mm −1 )

0.5

ft / f n Stationary time: 400s

0.4

0.3

0.2 0.0

0.01

0.02 0.03 ut (mm)

0.04

0.05

Fig. 15.12 Recovery of friction coefficient by the stop of sliding in the reduction process from the static-to kinetic-friction under the infinitesimal sliding velocity (Test data after Ferrero and Barrau, 1997)

374

15 Constitutive Equation for Friction

0.4

0.3

ft

max

/ fn 0.2 Experiment Present theory 0.1

0

10 20 30 40 Stationary contact time (s)

50

Fig. 15.13 Recovery process of friction coefficient with the elapsed time after the stop of sliding (Test data after Brockley and Davis, 1968)

The comparison with test data for the recovery of friction coefficient is shown in Fig. 5.13. The sliding was first given reaching the kinetic-friction and then the tangential contact traction was unloaded to zero. Further, after the stop of sliding for a while the sliding was given again. The relations of the maximum value of frictional coefficient ft max / f n vs. the time elapsed after the stop of sliding are plotted in this figure. The test curve for sliding between hardened and fully-annealed coppers (Brockley and Davis, 1968) is simulated sufficiently well by the present model, where the material parameters are selected as follows:

F0 = Fs = 120MPa, Fk = 200MPa, M = 0.16,

κ = 10MPa/mm, m = 2, ξ = 230MPa/s, n = 2, u = 3 000mm −1 ,

α n = α t = 10MPa/mm, f n = 138kPa, vt = 0.1mm/s (before stop of sliding) The similar comparison with test data for sliding between cast irons (Kato et al., 1972) is shown in Fig. 5.14. The test data are simulated quite well by the present model, where the material parameters are selected as follows:

F0 = Fs = 3 000kPa, Fk = 50kPa, M = 0.19,

κ = 5MPa/mm, m = 2, ξ = 20MPa/s, n = 2, u = 1 000mm −1 , α n = αt = 10MPa/mm, f n = 33.4kPa, vt = 0.05mm/s (before stop of sliding)

15.9 Extension to Orthotropic Anisotropy

375

0.5 0.4

ft

max

/ fn 0.3 0.2 0.1 0

Experiment Present theory 50 100 150 200 Stationary contact time (s)

250

Fig. 15.14 Recovery process of friction coefficient with the elapsed time after the stop of sliding (Test data after Kato et al., 1972)

15.9 Extension to Orthotropic Anisotropy The difference of friction coefficients in mutually opposite sliding directions can be described by the aforementioned rotational anisotropy. However, the difference of friction coefficients in the mutually-perpendicular directions cannot be described by the rotational anisotropy. In order to extend so as to describe this difference, let the concept of orthotropy be further incorporated below. The simple surface asperity model is illustrated in order to obtain the insight for the anisotropy in Fig. 15.15. Here, the directions in the inclination of surface asperities would lead to rotational anisotropy. In addition, the anisotropic shapes and intervals of surface asperities would lead to the orthotropic anisotropy. Now, choosing the bases e*1 and e*2 in the directions of the maximum and the minimum principal directions of anisotropy, respectively, and letting e*3 coincide with n to make the right-hand coordinate system (e1∗ , e∗2 , e∗3 ) , it can be written as

f = f1∗ e∗1 + f2∗e∗2 + f 3∗e3∗ ⎫⎪ ⎬ β = β 1∗ e1∗ +β 2∗e∗2 + β3∗e∗3 ⎪⎭

(15.86)

while the spin Ω of the base (e1∗ , e∗2 , e∗3 ) coinciding with that of the contact surface and thus it holds that

376

15 Constitutive Equation for Friction

Ω ≡ e• *r ⊗ e*r , e• ∗i = Ω e∗i

(15.87)

Equation (15.86) is rewritten by f1∗ = f t 1∗, f2∗ = f t 2∗, f3∗ = − f n∗, β1∗ = βt 1∗, β2∗ = β t 2∗, β3∗ = 0 as follows: f = f t 1∗e∗1 + f t 2∗e∗2 − fn e∗3 ⎫⎪ ⎬ β = βt∗1 e1∗ + βt∗2 e2∗ ⎪⎭

(15.88)

e∗2

e1∗

Fig. 15.15 Surface asperity model suggesting the rotational and the orthopic anisotropy

Invoking the orthotropic anisotropy proposed by Mroz and Stupkiewicz (1994), let Eq. (15.52) with Eq. (15.53) taking account of the rotational anisotropy be extended as follows: f (f , β ) = fn g ( χˆ ∗ )

fn g ( χˆ ∗ ) = F , fn g ( χˆ ∗ ) = RF

(15.89) (15.90)

15.9 Extension to Orthotropic Anisotropy

377

where

ˆ∗ ˆ∗ χˆ ∗ ≡ χˆ1∗2 + χˆ 2∗2 , χˆ1∗≡ η1 , χˆ2∗≡ η 2

(15.91)

M2

M1

M1 and M 2 are the material constants standing for the values of M in the maximum and the minimum principal directions of anisotropy, respectively. The sliding-yield surface having the rotational and the orthotropic anisotropy is depicted in Fig. 15.16.

Rota tio n

D

f n M2

f

ft fn M2

β 2∗

ȕ

0

β 1∗

D

f 2∗

v v2∗

D

f1∗

f 2∗

v1∗ f1∗

fn M 1

fn M 1

fn

Normal sliding yield surface

f Slidingsubloading surface

e∗3 = n

0

e∗2

e1∗

Fig. 15.16 Sliding-yield surface with the rotational and the orthotropic anisotropy

378

15 Constitutive Equation for Friction

The partial derivatives for Eq. (15.89) are given as

∂χˆ i∗ = ∂ ( f t∗i / f n − βi∗) / M i = 1 fn Mi ∂ f t∗i ∂ f t∗i

⎫ ⎪ ⎪ ⎪ − f t∗i χ ∗i ⎪ ∂χˆ i∗ ∂ ( f t∗i / f n − βi∗) /M i =− = = 2 ⎪ f ∂ fn ∂ fn fn M i n ⎪ ⎪⎪ ˆ ∗ χ ∂ = 1 2 χˆ i∗ = ζˆi∗ ⎬ ∗ ˆ 2 χˆ ∗ ∂χ i ⎪ ⎪ ˆ ∗ ∂χˆ ∗ ∂ χˆ ∗ ∂χˆi∗ χˆi∗ 1 1 ζi ⎪ = = = ∂χˆ i∗ ∂ f t∗i χˆ ∗ f n M i f n M i ⎪ ∂ f t∗i ⎪ ⎪ ∂χˆ ∗ ∂ χˆ ∗ ∂χˆ1∗ ∂ χˆ ∗ ∂χˆ2∗ + = − 1 (ζˆ1∗ χ1∗ + ζˆ2∗ χ 2∗) ⎪ = ∗ ∗ ˆ ˆ f f f χ χ f ∂ n ∂ 1 ∂ n ∂ 2 ∂ n n ⎪⎭

(15.92)

where

ζˆi ≡

χˆi∗ χˆ ∗

(15.93)

Subscript i takes 1 or 2 and is not summed even when it is repeated. It holds from Eqs. (15.21) and (15.92) that ∂χˆ ∗ ∂χˆ ∗ e∗ + ∂χˆ ∗ e∗2 − ∂χˆ ∗ n 1 ζˆ1∗ e∗ + ζˆ2∗ e∗ + (ζˆ ∗ χ ∗ + ζˆ ∗χ ∗) n 2 = ∗ 1 } = f { 1 2 1 2 1 ∂ fn ∂f ∂ f t1 ∂ f t∗2 n M1 M2

(15.94)

∂ ( fn g ( χˆ ∗)) ∂fn g ˆ ∗ χˆ ∗ = ( χ ) + fn g' ( χˆ ∗) ∂ ∂f ∂f ∂f ∗ ∗ ˆ ˆ ζ ζ = − g ( χˆ ∗)n + fn g' ( χˆ ∗) 1 { 1 e∗1 + 2 e∗2 + (ζˆ1∗χ1∗ + ζˆ2∗χ 2∗) n}

N=

f n M1

M2

ζˆ1∗ ∗ ζˆ2∗ ∗ e2 + (ζˆ1∗χ1∗ + ζˆ2∗χ 2∗) n} e1 + M1 M2

= − g ( χˆ ∗) n + g' ( χˆ ∗){

= g' ( χˆ ∗ )(

ζˆ1∗ ∗ ζˆ2∗ ∗ e1 + e2 ) − { g ( χˆ ∗ ) − g' ( χˆ ∗ )(ζˆ1∗ χ1∗ + ζˆ2∗ χ 2∗)}n M1

M2

(15.95)

15.9 Extension to Orthotropic Anisotropy

379

(I − n ⊗ n) • N = (I − n ⊗ n) •

ˆ∗

[g' ( χˆ ∗)(ζM

1

e∗1 +

1

ζˆ2∗ ∗ e2 ) − { g ( χˆ ∗ ) − g' ( χˆ ∗) (ζˆ1∗ χ1∗ + ζˆ2∗ χ 2∗)}n

]

M2

ζˆ ∗ ζˆ ∗ = g ' ( χˆ ∗ ) ( 1 e∗1 + 2 e∗2 ) M2 M1

(15.96)

ζˆ1∗ ∗ ζˆ2∗ ∗ e1 + e2 tn =

M1

M2

ζˆ∗

2

( ) +( 1

M1

(15.97)

ζˆ ∗ 2

M2

)

2

ζˆ∗ ζˆ∗ N • Ce = g' ( χˆ ∗ )( 1 e∗1 + 2 e∗2 ) −{ g ( χˆ ∗) − g' ( χˆ ∗)(ζˆ1∗χ1∗ + ζˆ2∗χ 2∗)}n M1 M2

[

]

•{α n e∗ 3 ⊗

= α t g' ( χˆ ∗ )(

e∗3 + α t (e∗1 ⊗ e∗1 + e∗2 ⊗ e∗2 )}

ζˆ1∗ ∗ ζˆ2∗ ∗ − α { g ( χˆ ∗ ) − g ( χˆ ∗ )(ζˆ∗ χ ∗ + ζˆ ∗ χ ∗)}n e1 + e2 ) n ' 2 1 2 1 M1

M2

(15.98)

ζˆ∗ ζˆ∗ N • Ce • t n = α t g' ( χˆ ∗)( 1 e∗1 + 2 e∗2 ) M1 M2

[

]

−α n{ g ( χˆ ∗) − g' ( χˆ ∗)(ζˆ1∗ χ1∗ + ζˆ∗2 χ 2∗)}n

ζˆ1∗ ∗ ζˆ∗ e1 + 2 e∗2

M1

(Mζ ∗ ) + (Mζ ∗ ) ˆ

1

1

= α t g' ( χˆ ∗ )

ˆ∗

ˆ∗

(ζ ) + ( ζ ) 1

M1

2

2

M2

2

2

ˆ

2

2

2

(15.99)

M2

Substituting Eqs. (15.21) and (15.95)-(15.99) into Eqs. (15.42) and (15.44), we obtain the sliding velocity vs. contact traction rate and its inverse relation as follows:

380

15 Constitutive Equation for Friction

D D D 1 1 v = {α n ⊗ n + α ( I − n ⊗ n)}( f 1 e∗1 + f 2 e∗2 − fn n) n t

[g' ( χˆ ∗)(ζMˆ∗ e∗ + Mζˆ∗ e∗ ) −{g ( χˆ ∗) − g' ( χˆ ∗)(ζˆ∗χ ∗ + ζˆ∗χ ∗)}n] 1

1

1

2

1

2

1

2

2

D D ∗ D c • ( f 1e∗ 1 + f 2 e2 − fn n) − m

2

+

mp

D D D = α1 ( f 1 e∗1 + f 2 e∗2 ) − α1 fn n n t ˆ∗ D ˆ∗ D g' ( χˆ ∗ ) ζ1 f 1 + ζ 2 f 2 M1 M2 χ ∗ D mc +{ g ( χˆ ∗ ) − g' ( χˆ ∗) (ζˆ1∗ χ1∗ + ζˆ∗ 2 2 ) fn − + mp

(

)

ζˆ1∗ ∗ ζˆ∗ e1 + 2 e∗2

M1

M2

(Mζ ∗ ) + (Mζ ∗ ) ˆ

1

ˆ

2

1

2

2

2

(15.100)

[

D f = {α n n ⊗ n + α t (e∗1 ⊗ e∗1 + e∗2 ⊗ e∗2 )}

ˆ∗

{αt g' ( χˆ ∗)( ζM

1

−

〈

v 1e∗1 + v 2e∗2 − vn n

e∗1 +

ζˆ2∗ ∗ − α { g ( χˆ ∗) − g ( χˆ ∗) e2 ) n '

M2 1 ∗ ˆ ˆ ∗ (ζ 1 χ1 + ζ 2∗χ 2∗)}n

}• (v e∗ + v e∗ − vn n ) − m c

m p + α t g' ( χˆ ∗)

1 1

ˆ∗

ˆ∗

1

2

M2

M2

M1

= α t (v 1 e∗1 + v2 e∗2 ) − α n vn n

2

2

M1

ˆ∗

(ζ ) + ( ζ ) 1

ˆ∗

(ζ ) + ( ζ )

ζˆ1∗ ∗ ζˆ2∗ ∗ e1 + e2 M1

2 2

2

M2

2

]

2

〉

15.9 Extension to Orthotropic Anisotropy

−α t

〈

381

ˆ∗ ˆ∗ α t g' ( χˆ ∗)( ζ 1 v 1 + ζ 2 v2 ) M1

M2

+ α n{ g ( χˆ ∗) − g' ( χˆ ∗) (ζˆ1∗χ1∗ + ζˆ2∗χ 2∗)}vn − m c m p + α t g' ( χˆ ∗)

ζˆ ∗ 1

M1

M1

ζˆ ∗ 2

M2

1

2

2

〉

M2

e∗2 (15.101)

ˆ∗

ζ ( ζM ) + ( M ) 1

ζˆ2∗

2

( ) +( )

e∗1 +

ˆ∗

ζˆ1∗

2

2

2

Equation (15.70) with Eq. (15.71) for the Coulomb friction condition with rotational anisotropy can be extended to the orthotropic anisotropy as follows: f (f , β ) = χˆ c∗ , F = μ

(15.102)

χˆ c∗ = μ , χˆ c∗ = R μ

(15.103)

where ∗ C1

∗ C2

χˆ c∗ ≡ χˆ c∗12 + χˆ c∗22 , χˆ c∗1 ≡ ηˆ1 , χˆ c∗2 ≡ ηˆ2

(15.104)

C1 = 1 and C2 ( ≤ 1) is the material constant, whereas the e∗ 1 - direction is chosen for the long axis of ellipsoid in the section of sliding-yield surface so that μ designates

the friction coefficient in the e∗ 1 - direction. The partial derivatives for Eq. (15.102) are given as follows: ∂ χˆ c∗i ∂ ( f t∗i / f n − βi∗) / C i = = 1 f n Ci ∂ f t∗i ∂ f t∗i

½ ° ° ° ° ˆ χˆ c∗i − f t∗i ∂ χc∗i ∂ ( f t∗i / f n − βi∗) /Ci ° − = = = fn ∂ fn ∂ fn fn2 Ci ° ° °° ∂ χˆ ∗ c = 1 2 χˆ c∗i = ζˆc∗i ¾ ˆc ° (15.105) ∂ χˆ c∗i 2χ ∗ ° χˆ ∗ χˆ ∗ χˆ ∗ ζˆ ∗ ° ∂ χˆ ∗ c = ∂ c ∂ ci = ci 1 = 1 ci ° f C ˆ f C ∗ ∗ χ∗ n i ∂ f ti c n i ∂ χˆc∗i ∂ f t i ° ° ˆ c ∂ χˆc∗i ∂ χˆ ∗ ∂ χˆc∗i ° ∂ χˆ ∗ c = ∂χ∗ c 1 ζˆc∗i χ c∗1 + ζˆc∗i χ c∗2 ° =− + f f f ∂ fn ∂ ∂ ˆ ˆ n n n χ χ ∗ ∗ ∂ ci ∂ ci °¿

(

)

382

15 Constitutive Equation for Friction

Further, it holds from Eqs. (15.21) and (15.105) that

N=

ˆ∗ ˆ∗ ∂ χˆ c∗ 1 ζ c 1 e∗ + ζ c 2 e∗2 + (ζˆ∗ χ ∗ + ζˆ∗ χ ∗ ) n} (15.106) 1 = f { c2 c c1 c1 C2 C1 n ∂f

ζˆ ∗ ζˆ ∗ (I − n ⊗ n) • N = (I − n ⊗ n) • 1 { c 1 e∗1 + c 2 e∗2 + (ζˆc∗1 χ c∗1 + ζˆc∗2 χc∗ ) n} f n C1 C2 ζˆc∗ ζˆ ∗ = 1 ( c 1 e∗1 + 2 e∗2 ) f n C1 C2

(15.107)

ζˆc∗1 ∗ ζˆc∗2 ∗ e1 + e C2 2 C1 tn = ˆ∗ ˆ∗ ( ζCc11 )2 + (ζCc22 )2

(15.108)

∗ ζ c∗2 1 ζ c1 ∗ ˆ∗ e N • C = f { C e∗1 + C e∗2 + (ζˆc∗1 χc 1 + ζ c 2 χc∗)n} n 1 2 •{α n e∗ 3 ⊗

e∗3 + αt (e∗1 ⊗ e∗1 + e∗2 ⊗ e∗2 )}

ζˆ∗ ζˆ∗ 1 = f α t ( c 1 e∗1 + c 2 C2 C1 n

{

) + α n (ζˆc∗ χc∗ + ζˆc∗ χc∗ ) n} 1

1

ζˆc∗ ζˆc∗ 1 N • Ce • tn = f α t ( 1 e∗1 + C 2 C1 n 2

2

(15.109)

) + α n (ζˆc∗ χc∗ + ζˆc∗ χc∗) n}

{

1

1

2

ζˆc∗1 ∗ ζˆc∗2 ∗ e1 + e2 C1 ζˆc∗

•

2

C2 ζˆc∗

(C ) +(C ) 1

2

1

=

αt fn

ˆ∗

ˆ∗

( ζCc ) + (ζCc ) 1

1

2

2

2

2

2

2

(15.110)

15.9 Extension to Orthotropic Anisotropy

383

Substituting Eqs. (15.21) and (15.106)-(15.110) into Eqs. (15.42) and (15.44), we obtain the sliding velocity vs. contact traction rate and its inverse relation as follows: D D D v = {α1 n ⊗ n + α1 ( I − n ⊗ n)} ( f 1 e∗1 + f 2 e∗2 − fn n) n

t

ˆ∗ ζˆ2c∗ ∗ ˆ∗χ c∗ + ˆ∗χ ∗ 1 ζ1c e∗ e + (ζ1c 1 ζ2c c )n + 1 f n C1 C2 2 D D ∗−D • ( f e∗ fn n) − m c 1 1 + f 2 e2

}

{

ζˆ1c∗ ∗ ζˆ2c∗ ∗ e + e C1 1 C2 2

mp

ˆ∗

ˆ∗

(Cζ c ) + (ζCc ) 2

1 1

2

2

2

D D D = α1 ( f 1 e∗1 + f 2 e∗2 ) − α1 fn n n t

+

ˆ∗ 1 ζ1cf f n C1

{

D ζˆ2c∗ D 1+

ˆ∗χ ∗ + ˆ∗χ ∗ fD f C2 2 − (ζ1c c1 ζ2c c ) n mp

} − mc

ζˆ1c∗ ∗ ζˆ2c∗ ∗ e1 + e2 C2

C1

ˆ∗

ˆ∗

(Cζc ) + (ζCc ) 1 1

2

2

2

2

(15.111)

D f = {α n n ⊗ n + α t (e∗1 ⊗ e∗1 + e∗2 ⊗ e∗2 )} v 1e∗1 + v2e∗2 − vn n

(

−

〈

ζˆ1c∗ ∗ ζˆ2c∗ 1 ˆ∗χ ∗ + ˆ ∗χ n f n α t C1 e1 + C2 + α n (ζ1c c1 ζ2c c∗) • (v 1e∗ 1 + v 2e∗ 2 − vn n ) − m c

{ (

}

)

α

t mp + f n

ˆ∗

ˆ∗

(Cζc ) + (ζCc ) 1 1

2

2

2

2

〉

ζˆc∗1 ∗ ζˆc∗2 ∗ e1 + e2 C2

C1

ˆ∗

1

1

= α t (v 1 e∗1 + v2 e∗2 ) − α n vn n

−

〈

∗ 1 α ( ζˆc∗1 v 1 + ζˆc 2 v ) − α n (ζˆ∗ χ c∗ + ζˆ ∗ χ ∗ ) vn} − m c c2 c t c1 1 fn C2 2 C1

{

m p + α t g' ( χˆ ∗ )

ζˆ1∗

2

( ) +( M1

ζˆ2∗ M2

)

2

ˆ∗

( ζCc ) + (ζCc )

〉

2

2

2

2

384

15 Constitutive Equation for Friction

ζˆc∗1 ∗ ζˆc∗2 ∗ e1 + e2 C2

C1

ζˆc∗

(15.112)

ζˆc∗

2

(C ) +( C ) 1

2

1

2

2

The calculation for sliding with the orthotropic anisotropy must be performed in the coordinate system with the principal axes of orthotropy, i.e. (e1∗ , e∗2 , n ) . We examine below the basic response of the present friction model by numerical experiments and comparison with test data for the linear sliding phenomenon without a normal sliding velocity leading to

vn = 0

(15.113)

The traction rate vs. sliding velocity relation is given by substituting Eq. (15.113) into Eq. (15.101) as D f = αt (v 1 e∗1 + v2 e∗2 )

− αt

〈

ˆ∗ ˆ∗ α t g' ( χˆ ∗)( ζ 1 v 1 + ζ 2 v2 ) − m c M1

m p + α t g' ( χˆ ∗)

M2 ζˆ1∗ 2

ζˆ ∗

M1

M2

( ) +( ) 2

2

〉

ζˆ1∗ ∗ ζˆ2∗ ∗ e1 + e2 M1 ζˆ∗

M2 ζˆ∗

2

(M ) + (M ) 1

2

1

(15.114) 2

2

while it holds that fn = const. The traction rate vs. sliding velocity relation is given by substituting Eq. (15.113) into Eq. (15.112) as D f = α t (v 1 e∗1 + v2 e∗2 )

−

〈

ζˆc∗2 1 α ζˆc∗1 − mc f n t ( C1 v1 + C2 v2 ) ˆ∗ 2 ˆ∗ 2 m p + αt g' ( χˆ ∗) (ζ 1 ) + (ζ 2 ) C1 C2

〉

ζˆc∗1 ∗ ζˆc∗2 ∗ e1 + e2 C2

C1

ζˆc∗

2

ζˆc∗2

(C ) +(C ) 1

1

where m p and m c are given by Eqs. (15.73) and (15.74).

2

(15.115) 2

15.9 Extension to Orthotropic Anisotropy

385

The influence of the rotational hardening is shown by the numerical experiments using Eq. (15.115) for the Coulomb friction condition in Fig. 15.17, where the material parameters are selected as follows:

μ 0 = μ s = 0.2, μ k = 0.2, M = 0.19, κ = 10/mm, m = 1, ξ = 0.1/s, n = 1, u = 1 000mm −1 , α n = α t = 1000MPa/mm, f n = 10MPa. assuming β2 = β3 = 0, vt 2 = vt 3 = 0 . It is shown that the difference of friction coefficients in the opposite sliding directions increases with the rotational anisotropy.

vt1 = ±0.001 vt 1 = ±0.1

vt 1 = ±0.0001 vt 1 = ±0.01 6 4

6

β t 1 = 0.05 μ s μk

2

ft 1 fn

ft 1 fn

μk

-2

-6 -0.2

0

-4

0

ut 1 (mm)

μk

μk

-2

μs -0.1

μs

2

0

-4

β t 1 = 0.1

4

0.1

0.2

-6 -0.2

μs -0.1

0

0.1

0.2

ut 1 (mm)

Fig. 15.17 Influence of rotational anisotropy on relation of tangential traction and displacement

Appendixes

Appendix 1: Projection of Area Consider the projection of the area having the unit normal vector n onto the surface having the normal vector m . Now, suppose the plane ( ,abcd in Fig. A.1) which contains the unit normal vectors m and n . Then, consider the line ef obtained by cutting the area having the unit normal vector n by this plane. Further, divide the area having the unit normal vector n to the narrow bands perpendicular to this line and their projections onto the surface having the normal vector m . The lengths of projected bands are same as the those of the original bands but the widths become to the ones multiplied by the scalar product of the unit normal vectors, i.e. m • n . Eventually, the projected area da is related to the original area da as follows:

da = m • n da

a e n

b db d

m

f

n • m db c Fig. A.1 Projection of area

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 387–393. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

(A.1)

388

Appendixes

Appendix 2: Proof of ∂ ( F jA / J ) / ∂ x j = 0 F jA J = 1 {∂ (∂ x j / ∂X A ) J − ∂ x j ∂ J } ∂X A ∂ x j J2 ∂x j ∂x j

∂

= 12 J

∂ (∂ x j / ∂X A ) ε ∂x1 ∂x2 ∂x3 − ∂ x j PQR ∂X P ∂X Q ∂X R ∂XA ∂x j

{

= 12 J

{( ∂X∂ x∂x + ∂X∂ x∂ x 2

2

1

A

A

1

2

2

+

∂ε PQR ∂x1 ∂x2 ∂x3 ∂X P ∂X Q ∂X R ∂x j

}

∂ 2 x3 ε ∂x1 ∂x2 ∂x3 ) ∂X A ∂ x3 PQR ∂X P ∂X Q ∂X R

2x 2x 2x x x x − ε PQR ( ∂ 1 ∂ 1 ∂x2 ∂x3 + ∂ 2 ∂x1 ∂ 2 ∂x3 + ∂ 3 ∂x1 ∂x2 ∂ 3 )} ∂XA ∂X P ∂ x1 ∂X Q ∂X R ∂XA ∂X P ∂X Q∂ x2 ∂X R ∂XA ∂X P ∂X Q ∂X R∂ x3

{( ∂X∂ x∂x ∂∂Xx 2

= 12 ε PQR J

A

1

1

1

P

∂x2 ∂x3 + ∂x1 ∂ 2 x2 ∂x2 ∂x3 + ∂x1 ∂x2 ∂ 2 x3 ∂x3 ) ∂X Q ∂X R ∂X P ∂X A ∂ x2 ∂X Q ∂X R ∂X P ∂X Q ∂X A ∂ x3 ∂X R

2x 2x 2x x x x − ( ∂ 1 ∂ 1 ∂x2 ∂x3 + ∂ 2 ∂ 2 ∂x1 ∂x3 + ∂x1 ∂x2 ∂ 3 ∂ 3 ∂X P ∂ x1 ∂XA ∂X Q ∂X R ∂X A ∂X Q∂ x2 ∂X P ∂X R ∂X P ∂X Q ∂X R∂ x3 ∂XA

)}

∂ 2 x3 ∂ 2 x1 ∂x2 ∂x3 ∂ 2 x2 ∂x3 ) = 12 ε PQR + ∂x1 + ∂x1 ∂x2 ∂X A ∂X P ∂X Q ∂X R ∂X P ∂X A ∂X Q ∂X R ∂X P ∂X Q ∂X A ∂X R J

{

2x 2x 2x − ( ∂ 1 ∂x2 ∂x3 + ∂x1 ∂ 2 ∂x3 + ∂x1 ∂x2 ∂ 3 ∂X P ∂X A ∂X Q ∂X R ∂X P ∂X Q ∂X A ∂X R ∂X P ∂X Q ∂X R ∂X A

)} = 0 (A.2)

Appendix 3: Euler’s Theorem for Homogeneous Function The homogeneous function of degree

f (ax1 , ax2 ,

• ••,

n is defined to fulfill the relation axm ) = a n f ( x1 , x2 ,

• ••,

xm )

(A.3)

assuming the variables x1 , x2 , •••, xm and letting a denote an arbitrary scalar constant. Then, the homogeneous function can be given by the polynomial expression: s

f ( x1 , x2 ,

• ••,

ni ni

ni

xm ) = ¦ x1 1 x2 2 • • • xmn i =1

(A.4)

Appendix 4: Normal Vector of Surface

where

389

s is the number of terms, provided to fulfill m

n ij = n for each i ¦ j

(A.5)

=1

Eq. (A.4) leads to s ∂f ( x1 , x2 , • ••, xm ) xj = ¦ ∂x j 1 i =1

m

¦ j=

m

nij x ¦ j=

ni1 n i2

1

x2

nij −1 ni •• • x j • • • xmn x j

1

s

ni ni

ni

= n¦ x1 1 x2 2 • • • xmn i =1

Then, it holds that m

∂f ( x1 , x2 , • ••, xm ) x j = n f ( x1 , x2 , ∂x j 1

∑ j=

• ••,

xm )

(A.6)

which is called the Euler’s theorem for homogeneous function. For the simple example ( m = 3, n = 4, s = 3 ):

f ( x, y, z ) = α x 4 + β x3 y + γ x 2 yz Eq. (A.6) is confirmed as follows: ∂f ∂f ∂f x + y y + z = (4α x3 + 3β x 2 y + 2γ xyz ) x + ( β x 3 + γ x 2 z ) y + γ x 2 y • z = 4 f ∂x ∂ ∂z

Eq. (A.6) leads to Eq. (6.32) for the yield function (n = 1) .

Appendix 4: Normal Vector of Surface The quantity tr{(∂ f (σ) / ∂σ)dσ} is regarded as the scalar product of the vectors ∂ f (σ )/ ∂σ

and dσ in the nine-dimensional space. Here, it holds that

∂f ( )d tr( ∂

)

> 0 : d is directed outward of yield surface = 0 : d is directed tangential to yield surface < 0 : d is directed inward of yield surface (A.7)

Therefore, ∂ f (σ )/ ∂σ is interpreted to be the vector designating the outward-normal of the yield surface. This fact holds also for general scalar function f (T ) of arbitrary tensor T .

390

Appendixes

Appendix 5: Convexity of Two-Dimensional Curve When the curve is described by the polar coordinates ( r , θ ) as shown in Fig. A.1, the following relation holds tan α = rdθ dr

(A.8)

where α is the angle measured from the radius vector to the tangent line in the anti-clockwise direction. Eq. (A.8) is rewritten as

cot α = rr' ,

(A.9)

where ( )' designates the first order differentiation with respect to θ .

Cu rv e

y

α

drr

R

dr

rd

rd e nt l i

rr α

ne

− ππ /2 Θ Θ

θ

Ta ng

d

θθ θθ

α

θθ

Θ − {θ − (π / 2 − α )} 0

x

Fig. A.2 Curve in the polar coordinate (r, θ )

The equation of the tangent line at ( r , θ ) of the curve r = r (θ ) is described by the following equation by using the current coordinates ( R , Θ ) on the tangent line.

Appendix 6: Derivation of Eq. (11.19)

391

R cos[Θ − {θ − (π / 2 − α )}] = rcos(π / 2 − α ) which is rewritten as − R sin(Θ − θ − α ) = rsinα →

1 =− 1 rsinα R cos(Θ − θ − α )

→ 1 = 1r cos(Θ − θ ) − 1r cot α sin(Θ − θ ) R Substituting Eq. (A.9) to this equation and noting (1/ r )' = −r'/r 2 , one has the relation 1 = 1 cos(Θ − θ ) + ( 1 )' sin(Θ − θ ) , r R r

(A.10)

Equation (A.10) is rewritten by the Taylor expansion as 1 = 1 cos ϑ + ( 1 )' sin ϑ = 1 (1 − 1 ϑ 2 + • • •) r (θ ) r (θ ) 2 R(Θ ) r (θ ) + ( 1 )' (ϑ − 1 ϑ 3 + • • •) r (θ ) 6 = 1 + ( 1 )'ϑ − 1 1 ϑ 2 + • • • r (θ ) r (θ ) 2 r (θ )

(A.11)

where ϑ ≡ Θ − θ . On the other hand, the radius r (Θ ) for Θ = θ + ϑ on the curve is described by

1 = 1 + ( 1 )' + 1 ( 1 )" 2 . r (Θ ) r (θ ) r (θ ) ϑ 2 r (θ ) ϑ + • • •

(A.12)

Eqs. (A.11) and (A.12) lead to . 1 − 1 1 1 ( 1 )" 2 r (Θ ) R(Θ ) = 2{r (θ ) + r (θ ) }ϑ + • • •

(A.13)

In order that the curve is convex ( r (Θ ) ≤ R(Θ ) ), the following inequality must hold from Eq. (A.13). (A.14) 1 + ( 1 )" > 0 r r

Appendix 6: Derivation of Eq. (11.19) Differentiation of Eq. (11.11) under the condition f (σ ) = const. leads to χ

χ

g (χ ) dp + pg' ( χ )( ∂ p dp + ∂ d || σ' || ) ∂ ∂|| σ' ||

||σ || = g (χ )dp + pg' ( χ )(− p 2 ' dp + p1 d ||σ' ||) = 0 M M

392

Appendixes

from it holds that d||σ' || dp =

||σ || g (χ ) − p ' g' (χ ) g (χ ) M −χ = M g' (χ ) 1 g (χ ) M '

(

)

(A.15)

Considering d ||σ' || / dp = 0 at χ = 1 in Eq. (A.15), one has Eq. (11.19).

Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State The numerical experiments of small cyclic tension uniaxial loading behavior near yield state of 1070 steel are depicted in Fig. A3 for u in Eq. (8.36) and Fig. A4 σ a (MPa)

800 600

sa 400

200 αa

0

0.005

0.01

0.015

εa

0

0.005

0.01

0.015

εa

1.0 R

0.5

Fig. A.3 Numerical experiment of small cyclic tension uniaxial loading behavior near yield state of 1070 steel

Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State

393

σa (MPa)

800 600

sa 400 200 αa

0

0.005

0.01

0.015

εa

0

0.005

0.01

0.015

εa

1.0 R

0.5

Fig. A.4 Numerical experiment of small cyclic tension uniaxial loading behavior near yield state of 1070 steel for u = const.

for u = const. , respectively, while the same values of material parameters for Fig. 10. 7 are used (Hashiguchi and T. Ozaki, 2009). It is also here shown that the strain accumulation is suppressed by the extension of the material parameter u .

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Index

A

B

Accumulated plastic strain 146 Acoustic tensor 334 Additive decomposition of strain rate 142 Admissible field kinematically 108 statically 108 Admissible transformation 59 Almansi (Euler) strain 64, 82, 85, 113 Alternating symbol 2 Angular momentum 102 Anisotropy 156 kinematic 157, 186, 197, 222, 313, 324, 329 orthotropic 238 rotational 157, 197, 261 rotational for friction 354 sliding-yield surface 354, 375 traverse 317 yield surface 157, 186, 197, 222, 261, 313, 324, 329 Anti-symmetric tensor 23 Associated flow rule (Associativity) 145, 151, 158, 179, 203, 219 Drucker’s interpretation 151 plastic relaxation work rate 152 Prager’s interpretation 151 Associative law of vector 8 Axial vector 25

Back stress 157 Bauschinger effect 157 Bingham model 211 Biot strain tensor 67 stress tensor 125 Body force 102 Bounding surface 184 model 184 surface model with radial mapping 184 Bulk modulus 132, 251

C Caley-Hamilton theorem 35 Cap model 282 Cam-clay model 282 modified 256 original 256 Cartesian summation convention 1 decomposition 24 Cauchy ’s fundamental theorem 103 ’s law of motion 106 ’s stress principle 103 stress 103, 114, 128 elastic material 130 Cauchy-Green deformation tensor 63

408 Characteristic direction 26 equation 26 length 328 value 26 vector 26 Circle of relative velocity 80 Coaxility 35, 160 Cofactor 4 Commutative law of vector 8 Compliance method 336 Component of tensor 17, 20 vecto 10 Cofactor 4 Configuration current 57 initial 57 intermediate 136 reference 57 relative 58 Conservation law of angular momentum 102 mass 101 momentum 101 Consistency condition for conventional plasticity 143 extended subloading surface model 203 friction 357 initial subloading surface model 179 subloadong-superyield surface model 294, 299 Consolidation e − ln p linear relation 251 isotropic 249 ln v − ln p linear relation 249 normal 249 Contact elastic modulus 353 normal traction 352 tangential traction 352 traction 352 Continuity condition 171

Index Continuity equation 101 Continuum spin 71, 114, 118, 310, 313 Contraction of tensor 16 Convected stress 105 stress rate 123 Convective term 51 Conventional plasticity mode 135, 142 friction model 349 Convexity condition 254, 390 Convexity of yield surface 151, 254 Coordinate system 11 transformation 11, 15 Corner theory 160 Corotational rate 118 derivative 114, 119 Green-Naghdi (Dienes) 119, 122, 311, 316 Jaumann 107, 118, 122, 129, 168, 309, 315, 322 rate 110, 114, 118, 119, 309, 321 stress rate 122, 123, 309, 311, 315, 316 Cosserat elastic material 133 Cotter-Rivlin stress rate 123 Coulomb friction (sliding-yield) condition (surface) 350, 354, 360, 381 -Mohr failure criterion 285 Couple stress 133 Creep model 214 hardening of friction 356 Critical state 254 Cross product 9 Curl (rotation) of tensor field 52 Current configuration 57 Cyclic loading 170, 191 mobility 279, 295 plasticity m1odel 191 stagnation of isotropic hardening 225

Index

D Definition of tensor 14 vector 8 Deformation 57 gradient 59, 63, 136 of crystal 319 rate tensor 71 relative gradient 63 theory 161 Derivative material-time 50 spatial-time (local) 50 Description Eulerian 58 Lagrangian 57 material 57 relative 50, 58 spatial 50, 58 total Lagrangian 58 updated Lagrangian 58 Determinant 3 Vandermonde’s 39 Deviatoric part 24 invariant 30 plane ( π -plane) 41 projection tensor 162 tangential stress rate 162 tangential projection tensor 163 Dienes rate 119, 123, 311, 316 Differential calculi of tensor 48 Dilatancy locking 301 Direct (Symbolic) notation (description) 17 Direction cosine 10 Discontinuity of velocity gradient 333 Displacement 65 Dissipation energy 127 Distributive law of vector 8 Divergence of tensor field 51 Divergence theorem 53 Drucker’s postulate for stress cycle 151 classification of plasticity model 135

409 Drucker-Prager model 282 Dummy index 1 Dyad (tensor or cross product) 19 Dynamic loading ratio 218 surface 213

E Eddington’s epsiron 2 Eigen direction 26 projection 32 value 26, 31, 33 vector 26, 32, 33 Einstein’s summation convention 1 Elastic bulk modulus 132 constitutive equation 127 deformation gradient 136 material 127 modulus 131 predictor 341 shear modulus 131 strain energy 127 stress rate 153 volumetric strain 138 Elastic-plastic transition 182 Elastoplastic modulus tensor 145 Element test 327 e − ln p linear relation 252 Equilibrium equation 105 incremental(rate)-type 106 moment 107 Equivalent plastic strain 146 stress 146 viscoplastic strain 213 Euler ’s first law of motion 102 ’s second law of motion 102 ’s theorem for homogeneous function 142, 179, 202, 388 Eulerian description 58

410

Index

spin tensor 73, 90 strain tensor 65, 82, 85, 113 triad 61, 73, 86 Extended subloading surface model 192, 196, 265, 296

Hyperelasticity 127 Hyperelastic-plasticity 165 Hypoelasticity 131, 142, 309 Hypoplasticity 133, 160 Hysteresis loop 195

F

I

Failure surface 285 Finite strain theory 165 First Piola-Kirchhoff stress 103, 114, 127 Flow rule associated 145, 151, 158, 179, 203, 219 friction 358 nonassociated 284 plastic 143 viscoplastic 219 Footing settlement analysis 301 Frame-indiffrence 111 Functional determinant 59

Identity tensor fourth-order 21 second-order 21 Il-posedness 328 Ilyushin’s isotropic stress space 245 postulate of strain cycle 152 Impact load 211, 215, 220 Infinite surface model 195 Infinitesimal strain 65 Initial configuration 57 Initial subloading surface model 174, 281 Inner product 8 Intermediate configuration 136 Internal variable 143 Intersection of yield surfaces 160 Invariant 30 Inverse loading 206 relation 144, 168, 180, 205, 216 tensor 22 Isotropic consolidation 249 function 37 hardening (variable) 142, 222, 259 material 156 scalar-valued tensor function 28 tensor-valued tensor function 37

G Gauss’ s divergence theorem 53 Gradient of tensor field 51 Gradient theory 328 Green elastic material 127 strain 65, 82, 85, 113, 165 Green-Naghdi rate 119, 123, 311, 316

H Hamilton operator 51 Hardening 149, 180 anisotropic 157, 186 function 142, 222, 259 isotropic 142, 180, 222, 259 kinematic 157, 186, 222, 313 nonlinear-kinematic 195, 223 rotational 157, 186, 261 variable 142, 157 Hencky deformation theory 161 strain 69 Hooke’s law 131

J Jacobian 59 Jaumann rate 118, 309, 315 of Almansi strain tensor 78 of Cauchy stress 107, 122, 129, 168, 322 of Kirchhoff stress 123, 168 J2-deformation theory 161

Index

411

K

M

Kinematic hardening 157 linear 222, 313 nonlinear 195, 223 Prager 222 Ziegler 222 Kimatically admissible velocity field 108 Kinetic friction 349, 356 Kirchhoff stress 103, 114, 128 Kronecker’ delta 1

Macauley bracket 178 Magnitude of tensor 22 vector 8, 10 Masing rule 192, 206 Material description 57 frame-indifference 111 spin 118 -time-derivative 50, 55 Matrix 2 Maxwell model 211 Mean (Spherical) part 24 Mechanical ratcheting effect 193, 195, 233 Mesh size dependence (sensitivity) 328 Metal 146, 161, 221 Mises ellipse 246 yield condition 146, 244 Modified Cam-clay model 256 Modified overstress model 216 Mohr’s circle 45 Momentum 101 Motion 57 Multi surface (Mroz) model 182, 191 Multiplicative (Lee) decomposition of deformation gradient 135

L Lagrangian description 57 spin tensor 72, 93 strain 65, 82, 85, 113, 165 triad 61, 72, 87 Lame coefficients 131 Laplacian (Laplace operator) 52 Lee decomposition 136 Leftt Cauchy-Green deformation tensor 63, 114 polar decomposition 37, 60 relative Cauchy-Green deformation tensor 64 stretch tensor 61 Lie derivative 123, 167 Linear kinematic hardening rule 222, 313 transformation 16 ln v − ln p linear relation 249 Local form 106 Local theory 328 Local time derivative 50 Loading criterion for plastic sliding velocity 359 plastic strain rate 148, 180 Localization 327 Lode’s variable 44 Logarithmic strain 69, 79 volumetric strain 69, 139, 249

N Nabra 51 Nanson’s formula 98 Natural strain 79 Navier’s equation 130 Negative transformation 59 Nominal strain 79 stress 103, 114, 128 stress rate 107 stress vector 103 Nonassociated flow rule (Nonassociativity) 284

412 Nonhardening region 225 Nonlinear kinematic hardening model 182, 195 rule 223 Non-local theory 328 Non-proportional loading 160 Non-singular tensor 22 Non-steady term 50 Normal component 20 Normal isotropic hardening ratio 226, 229 surface 226 Normal stress rate 160, 162 Normal vector of surface 389 Normal sliding-yield ratio 355 surface 354 Normal-yield ratio 175, 197, 232, 271 surface 174, 197 Normality rule 145, 151 Normalized orthonormal base 9 Notation of tensor 17 Numerical calculation 340

O Objectivity 111 Objective stress rate tensor 122 tensor 15 transformation 15 Octahedral plane 147 shear stress 147 Oldroyd stress rate 123 Original Cam-clay model 256 Orthogonal tensor 18 Orthotropic anisotropy 238 Orthotropic anisotropy of friction 375 Overstress model 211

P Parallelogram law 8 Partial differential calculi 47

Index Perfectly-plastic material 149, 172 Permutation symbol 1 Perzyna’s over stress model 213 Piola-Kirchhoff stress 103 π -plane 42 Plane strain 248 stress 245 Plastic compressibility 249 corrector 342 deformation gradient 136 flow rule 143, 145, 151, 179, 203 modulus 144, 145, 179, 204, 230, 266, 281, 295, 300, 322 potential 145, 153 relaxation modulus tensor 144 relaxation stress rate 145, 153 shakedown 193 sliding flow rule 358 spin 317 strain rate 142, 144, 179, 203 volumetric strain 138 Poisson’s ratio 132 Polar decomposition 36, 60, 141 spin 72, 88, 311 Pole 47, 81 Positive definite tensor 35 proportionality factor 143-145, 159, 179, 180, 203 Potential energy 127 Prager’s continuity condition 172 kinematic hardening rule 222 overstress model 212 Prandtl-Reuss equation 147 Pressure-dependence 249 Principal direction 26 invariant 30 space 40 stretch 61

Index value 26 vector 26 Principle of maximum plastic work 155 objectivity 111, 141, 354 material-frame indifference 111 Product law of determinant 5 Projection of area 387 Projection tensor deviatoric 162 deviatoric-tangential 163 Pull-back 123, 165 Pulsating loading 233 Push-forward 123

Q Quasi-static deformation 215 Quotient law 15

R Ratcheting effect 193, 195, 233 Rate-dependence 211 Rate of area 99 extension 76 normal vector of surface 99 shear strain 76 volume 99 Rate-type equilibrium equation 106 virtual work principle 109 Reference configuration 57 Relative configuration 58 deformation gradient 63 description 58 left Cauchy-Green tensor 64 right Cauchy-Green tensor 64 spin 72, 88, 311, 313 Representation theorem 40 Return-mapping 340 Reynolds’ transportation theorem 56 Right Cauchy-Green deformation tensor 63, 114

413 polar decomposition 37, 60 relative Caucgy-Green deformation tensor 64 stretch tensor 61 Rigid-body rotation 111, 112 Rigid-plasticity 313, 317 Rotation (curl) of tensor field 52 Rotation of triad 61, 90, 93 Rotation rate tensor 71 Rotational hardening (variable) 157, 261 Rotational anisotropy of friction 354 Rotational strain tensor 133

S Second Piola-Kirchhoff stress 104, 114, 128 Scalar product (inner product) 8 triple product 9 Shakedown 193 Shear band 327 band inception 333 band thickness 331 -band embedded model 331 component 20 modulus 131, 251 strain rate 76 Similar tensor 28 Similarity-center 174, 197 surface 207 yield ratio 207 Similarity-ratio 174 Simple shear 84, 309 Single surface model 195 Skew-symmetric tensor 23, 33 Slidinghardening function 355 subloading surface 354 velocity 350 yield condition (surface) 354, 360 Smeared crack model 331 Smoothness condition 172 Softening 149, 180

414 Softening of sliding-yield surface 356 Soil 249 Spatial description 58 Spectral representation 31 Spherical part 24 Spin continuum 71, 114, 118, 310, 313 Eulerian 72, 90 Lagrangian 72, 93 of base 113 polar (relative) 72, 88, 311, 313 tensor 26 vector 26, 75 Stagnation of isotropic hardening 225 Static friction 350, 356 Statically-admissible stress field 108 Steady term 51 Strain 64 Almansi (Eulerian) 65, 82, 85, 113 Boit 67 cycle 153 energy function 127 Green (Lagrangian) 65, 82, 85, 113, 165 Hencky (logarithmic) 69 infinitesimal 64 logarithmic (natural) 69, 79 rate 71 space 150 tensor 64 Strain rate 70, 114 elastic 142 plastic 142, 144, 179, 203 viscoplastic 213, 219 Stress-controlling function 182 Stress cycle 151 Stress tensor 102 Biot 125 Cauhy 103, 114, 128 convected 105 Kirchhoff 103, 114, 178 first Piola-kirchhoff (nominal) 103, 114, 127, 219 nominal 103, 114, 128 second Piola-kirchhoff 104, 114, 128

Index Stress rate tensor 122 Convected (Cotter-Rivlin) 123 Jaumann of Cauchy stress 107, 122, 129, 168, 322 Jaumann rate of Kirhhoff stess 123, 168 Green-Naghdi (Dienes) 119, 123, 311, 316 nominal 107 Oldroyd 122 Truesdell 123 Truesdell rate of Kirhhoff stress 123, 129, 168 Stress vector (Traction) 101, 103, 352 Stretch left 61 principal 61 right 61 Stretching 71 Structure ratio 291 Subloading surface model 174, 196 normal-yield ratio 175, 197 normal-yield surface 174, 197 subloading surface 174, 197 stress-controlling function 182 Subloading overstress model 217 dynamic-loading ratio 218 dynamic-loading surface 218 subloading ratio 218 Subloading-friction model 350 sliding-hardening function 355 normal sliding-yield ratio 355, 356 sliding-subloading surface 355 normal sliding-yield surface 354 Sub-isotropic hardening stagnation surface model 225 normal-isotropic hardening ratio 226 normal-isotropic hardening surface 225 sub-isotropic hardening surface 226 Subloading-superyield surface model 291 substructure ratio 291 super-yield ratio 291 super-yield surface 291 Summation convention 1 Superposition of rigid-body rotation 112 Surface element 97

Index Symbolic notation (description) 17 Symmetric tensor 23 Skew-symmetric tensor 23, 33 Sylvester’s formula 33 Symmetry of Cauchy stress 108

T Tangent (stiffness) modulus 131 Tangential associated flow rule of friction 358 contact traction 352 inelastic strain rate 159, 187, 208, 220 stress rate 160, 162 velocity 351 Tension cut of yield surface 285 Tensor 14 acoustic 334 anti-symmetric 23, 33 Cartesian decomposition 24 characteristic equation 26, 29-31 coaxiality 35, 160 component 20 component notation 17 component notation with base 17 contraction 16 definition 14 deviatoric part 24 direct notation 17 eigenprojection 32 eigenvalue 26, 31, 33 eigenvector 26, 32, 33 identity 21 invariant 30 inverse 22 magnitude 22 matrix notation 17 mean (spherical) part 24 non-singular 22 normal component 20 notation 17 objective (transformation) 15 orthogonal 18 partial derivative 47 positive definite 35

415 principal direction 26 principal invariant 30 principal value 26 product (cross product, dyad) 19 polar decomposition 36, 60, 141 representation in principal space 40 shear component 20 similar 28 skew-symmetric 23, 33 similar 28 spectral representation 31 spin 26, 70, 72, 88, 113, 311 strain 64, 65, 67, 69, 79 strain rate 70, 114, 142, 179, 203, 213, 219 stress 102-105 stress rate 122-125 symbolic notation 17 symmetric 23 time-derivative 50 trace 20 transpose 21 triple decomposition 24 two-dimensional state 44 two-point 113, 136 unit 22 zero 21 Time-dependence 211 Time derivative local (spatial-time) 50 material-time 50 non-steady (local time derivative) term 50 steady (covective) term 51 Total Lagrangian description 58 Trace 20 Traction (Stress vector) 101, 352 Triad Eulerian 61, 93 Lagrangian 61, 72, 87 Transportation theorem 56 Transpose 3, 21 Traverse anisotropy 317 Triple decomposition 24 Truesdell stress rate 123 Truesdell rate of Kirchhoff stress 123, 129, 168

416 Two-dimensional state 44 Two-point tensor 113 Two surface model 182, 194

U Unconventional (elasto)plasticity 135, 171 friction model 349 Uniaxial loading 81, 233, 337 Uniqueness condition 171 Uniqueness of solution 171 Unit tensor 21 vector 8 Updated Lagrangian description 58

V Vandermonde’s determinant 39 Vector 8 associative law 8 axial 25 component description 9 commutative law 8 definition 8 direction 8, 10 distributive law 8 eigen (principal) 26 equivalence 8 magnitude 8 parallelogram law 8 product 9 scalar product 8 scalar triple product 9 spin 75

Index unit 8 zero 8 Velocity gradient 70, 336 Vertex of yield surface 160 Virtual work principle 108 Viscoelastic model 211 Viscoplastic coefficient 213 model 212 strain rate 213 Volume element 97 Volumetric strain (rate) 79, 140, 249 Vorticity 75

W Work Conjugacy 124 conjugate pair 125 hardening 147 rate 124, 152

Y Yield condition (surface) 142, 146, 150, 152, 238, 253 condition (surface) for friction 360, 363, 376, 381 Young’s modulus 132

Z Ziegler’s kinematic hardening rule 222 Zero tensor 21 vector 8